Proceedings
of the Tomsk Technological Institute
of Emperor Nicholas II.

Vol. 12. 1908. No. 4.

III.

V. L. Mal’tev,

Graphical Construction of Thermal Diagrams of Internal-Combustion Machines.

With 1 appended table.

1–24.

Graphical Construction of Thermal Diagrams of Internal-Combustion Engines

V. L. Maleev.

The indicator diagram, which is ordinarily used in studying the operation of internal-combustion engines, gives a clear picture of the phenomena occurring in the cylinder of the engine, but only from one point of view—the variation of the pressures in the cylinder at different moments of the piston position.

For a correct assessment of the operation of the engine it is very important to penetrate more deeply into the essence of the phenomena occurring in the cylinder and to obtain an idea of them from the point of view of thermal changes, i.e., to obtain a picture of the variation of temperature and heat. This picture will be obtained if the working process is represented graphically in such coordinates in which area represents heat, just as in \(PV\) coordinates (the indicator diagram) area represents work.

In these new coordinates, for an infinitely small change in the state of the working body (gas) from the state expressed by point \(1\) to state \(2\), Fig. 1, the infinitely small area \(122'1'1\) will consequently represent, on some scale, the change in heat that has occurred, \(dQ\), i.e.

\[ dQ = y\,dx. \tag{1} \]

Fig. 1.

One of the coordinates may be chosen arbitrarily; namely, let us take the absolute temperatures \(T\) as ordinates; then

\[ dQ = T\,dx, \]

whence

\[ x = \int \frac{dQ}{T}. \tag{2} \]

The expression \(\displaystyle \int \frac{dQ}{T}\), following Clausius’s proposal, is customarily called the entropy and denoted by \(S\). Thus

\[ dQ = T\,dS. \tag{3} \]

Comparing Eq. (3) with the general differential expression for heat,

\[ dQ=c\,dT, \tag{4} \]

where \(c\) is the heat capacity of the given body, one may conclude that the new concept—entropy \(S\)—must also be assigned to the order of heat capacities and is expressed in the same units, i.e. in heat units (calories).

The numerical value for \(S\) is easy to find by comparing Eq. (3) with the first law of thermodynamics

\[ dQ=dU+APdv; \tag{5} \]

we note that for a permanent gas, according to Joule’s law, the change in internal energy is

\[ dU=c_v dT, \tag{6} \]

where \(c_v\) is the heat capacity at constant volume; further, we replace, in the second term, the pressure \(P\) by its expression from the characteristic equation of the gas

\[ Pv=RT; \tag{7} \]

after these substitutions Eq. (5) takes the form

\[ dQ=c_v dT+\frac{ARTdv}{v}, \]

or, dividing both sides by \(T\) and noting that, by Eq. (3), \(\frac{dQ}{T}=dS\), we obtain

\[ dS=c_v \frac{dT}{T}+AR\frac{dv}{v}. \tag{8} \]

If the heat capacity \(c_v\) is regarded as a constant quantity, then both the first and the second term on the right-hand side of Eq. (8) are complete differentials; after integration we obtain

\[ S=c_v \, lg_n T+ARlg_n v+const. \tag{9} \]

We note that the product \(AR\) is equal to the difference of the heat capacities at constant pressure and constant volume, i.e.

\[ AR=c_p-c_v; \tag{10} \]

substituting this value of \(AR\) into Eq. (9), we obtain

\[ S=c_v \, lg_n T+(c_p-c_v)\,lg_n v+const.; \]

since \(lg_n T\) and \(lg_n v\) are dimensionless quantities, this equation once again clearly confirms that \(S\) belongs to the order of heat capacities.

In Eq. (9), \(S=f(T,v)\); the characteristic equation of the gas (7) contains three variable quantities, \(P\), \(v\), and \(T\); using Eq. (7), the expression for \(S\) may be represented in the form of a function of any pair of these three quantities, i.e. also in the form \(S=f(P,v)\) and, finally, \(S=f(P,T)\).

By simple elimination we obtain from Eq. (9)

\[ S=c_v \lg_n P+c_p \lg_n v+const., \tag{11} \]

and

\[ S=c_p \lg_n T-AR \lg_n P+const. \tag{12} \]

Of course, in all three expressions (9), (11), and (12) the constant quantity \(const.\) is different. Until we determine these constants, we shall not know the true, i.e. absolute, magnitude of \(S\), and at the same time we shall not know where the origin of the abscissa axis is in Fig. 1. But since, in the study of working processes, we are interested exclusively only in the successive changes of state and in the changes of \(S\) occurring thereby, i.e. only in the differences of the entropies of the initial and subsequent states, the origin of the coordinate axes along the \(S\)-axis may be chosen arbitrarily by assigning some value to \(const.\); for example, one may put \(const.=0\).

If we now have the indicator diagram of an internal-combustion engine, and, of course, its pressure scale \(m_p\) and volume scale \(m_v\) are known, and further know the volume of the compression chamber \(V_c\), the composition of the working gas (for example, from an analysis of the products of combustion), and the temperature at some one point of the diagram, for example at the beginning of compression, then we have all the data for transferring the working diagram from the \(PV\) coordinates into the \(TS\) coordinates. Namely, using Eq. (11), it is easy to calculate the values of \(S\) for a consecutive series of points of the indicator diagram, and, from the known temperature at some one point of the diagram, to find, by means of Eq. (7), the temperatures at the corresponding points. In this way we obtain, in the \(TS\) axes, a series of points which, joined by a continuous curve, will give a closed contour, i.e. a thermal or, as it is otherwise called, entropy diagram, the area of which represents the heat transformed into work in the given cycle.

However, such a method is very laborious and requires much time. This explains why the thermal diagram, despite the fact that it gives very valuable, one might say indispensable, indications in the investigation of the operation of existing internal-combustion engines, has not hitherto been used, or has hardly been used, not only in practice, but even in scientific laboratory investigations. In view of this, the graphical construction of the thermal diagram deserves great attention: a graphical method\(^1\) which not only considerably simplifies the transfer of the indicator diagram into the \(TS\) coordinates, but also requires incomparably less time.

\(^1\) The idea of this method was first indicated in the article by H. Mottet: “Le diagramme entropique appliqué à l’étude des cycles effectués par l’intermédiaire des gaz permanents.” — Annales de l’Association des ingénieurs de Gand, 1907.

We shall base this method on the following: let us take the expression \(S\) in the form of equation (11), setting \(const.=0\),

\[ S=c_v\,lg_n P+c_p\,lg_n v; \tag{11a} \]

it is not difficult to notice that each of the terms on the right-hand side of equation (11a) represents a linear quantity—the ordinate of a certain logarithmic curve. In order to find \(S\) for some point \(a\), for which \(P\) and \(v\) are known, for example given graphically in the coordinates \(PV\), Fig. 2, we draw with a cross four coordinate axes \(T\), \(S\), \(P\), and \(V\). Then, taking the axis \(OP\) as the axis of abscissae and the axis \(OT\) as the axis of ordinates, we construct, calculating point by point and choosing a scale for \(c_v\), the logarithmic curve \(c_v\,lg_n P\). Evidently, at \(P=S\) this curve will intersect the axis \(OP\), and further, for \(P<1\), its ordinates will be negative. Next, taking the axis \(OV\) as the axis of abscissae and the axis \(OS\) as the axis of ordinates, we construct in the same way the curve \(c_p\,lg_n v\), preserving for \(c_p\) the same scale that we chose above for \(c_v\). As before, for \(V<1\) the ordinates of this curve will be negative.

Fig. 2.

With the aid of these curves, the magnitude \(S_1\) for the point \(a\) is found as the sum of their ordinates \(cb\) and \(ed\). Graphically this addition is performed as follows: from point \(b\) draw a straight line \(\parallel\) to the axis \(OS\), and from point \(d\) a straight line \(\parallel\) to the axis \(OT\), up to their mutual intersection at point \(g\); with radius \(fg\), from point \(f\) cut off on the axis \(OS\) the line \(fh\);

\[ fh+Of=Oh=c_v\,lg_n P_1+c_p\,lg_n v_1=S_1. \]

Here, evidently, the scale of \(S\) will be the same as that chosen earlier for \(c_v\) and \(c_p\).

The second coordinate—the temperature \(T_1\)—is also found graphically, noting that, by equation (7), \(T_1\) is a fourth proportional to the quantities \(P_1\), \(v\), and \(R\). We choose for \(R\) the scale \(m_R\) and set off on it \(Oi=R m_R\), Fig. 2; we join \(i\) with \(c\) and draw \(ek \parallel ic\); then \(Ok=T_1\). The scale \(m_t\) for temperatures will evidently be

\[ m_t=\frac{m_p\cdot m_v}{m_R}. \tag{13} \]

Now we draw an arc of radius \(Ok\) to its intersection with the axis \(OT\) at the point \(j\); drawing through the points \(j\) and \(h\) straight lines respectively parallel to the coordinate axes, we obtain at their intersection the required point \(A\), corresponding in the \(TS\) axes to the point \(a\) given in the \(PV\) axes.

It may happen that \(V_2<1\), point \(a_2\), Fig. 2. Then \(c_p \lg_n v\) will be negative, but the entire course of the construction, as is seen from Fig. 2, is not changed by this.

All the preceding equations (1)–(12) were written for 1 kg of gas; if there are \(G\) kg of gas, then equation (7) is transformed into

\[ PV=GRT, \tag{14} \]

where \(V=Gv\). Instead of the quantity \(R\) in Fig. 2 one must set off, on the scale \(m_R\), the quantity \(GR\). As for the logarithmic curves, instead of \(c_v \lg_n P\) and \(c_p \lg_n v\) one must take \(Gc_v \lg_n P\) and \(Gc_p \lg_n v\), i.e. they would have to be reconstructed, increasing the ordinates \(G\) times. But since the scale for \(c_p\), \(c_v\), and \(S\) is in no connection with the scales \(m_p\), \(m_v\), \(m_t\), and \(m_R\), it is much simpler, without reconstructing the curves, to regard their scale as diminished \(G\) times.

It goes without saying that instead of equation (11) one may use the other two equations (9) and (12); the method remains exactly the same.

Let us take equation (9), putting \(\mathrm{const.}=0\),

\[ S=c_v \lg_n T+AR \lg_n v. \tag{9a} \]

Bearing equation (13) in mind, we choose the scale \(m_t\) for temperatures and construct, as before, by points, the curve \(c_v \lg_n T\), taking the axis \(OP\) as the axis of abscissas and the axis \(OT\) as the axis of ordinates, Fig. 3, and the curve \(AR \lg_n v\), taking as the axis of abscissas the axis \(OV\), and as the axis of ordinates—the axis \(OS\).

To transfer a point given in the \(PV\) coordinates, Fig. 3, into the \(TS\) coordinates, we proceed as follows: we draw the line \(ad\) to its intersection with the curve \(AR \lg_n v\); then \(ed=Of=AR \lg_n v_1\). Further, by the procedure described above, we find the temperature \(T_1\) at the point \(a\), having set off \(Oi=m_RR\),

in the form of the segment \(Ok\). The ordinate

\[ kj=c_v\,\lg_n T. \]

Carrying out, as in the preceding case, the graphical addition of ordinates, we obtain

\[ S_1=Of+fh=Oh; \]

then with radius \(Ok=T_1\), from the center \(O\), we draw an arc until it intersects the axis \(OT\) at the point \(j\); and at the intersection of the straight lines drawn from the points \(j\) and \(h\) parallel to the corresponding coordinate axes, we obtain the required point \(A\).

Fig. 3.

Finally, in Fig. 4 an example is shown of transferring the point \(a\) from the axes \(PV\) to the axes \(TS\) when using equation (12), where, as in the preceding cases, it is assumed that \(\mathrm{const.}=O\). To construct the curve \(c_p\lg_n T\), the axis \(OT\) must be taken as the axis of abscissas and the axis \(OS\) as the axis of ordinates; for the curve \(AR\lg_n P\), the axis \(OP\) must be taken as the axis of abscissas and the axis \(OT\) as the axis of ordinates. The transfer itself is carried out similarly to what has been described above; it is only necessary to remember that in this case the entropy \(S_1\) is equal not to the sum, but to the difference of the ordinates \(jm\) and \(jc\),

\[ S_1=Of-fh=Oh. \]

Some properties of logarithmic curves. Before proceeding to the discussion of the transfer of entire diagrams from the axes \(PV\) to the axes \(TS\), it is useful to recall some general properties of logarithmic curves.

If, as we did above and shall do subsequently, we assume that the heat capacities \(c_p\) and \(c_v\) are constant quantities (their difference \(c_p-c_v=AR\) is always constant), then the subtangent to logarithmic curves—

are, in equations (9), (11), and (12), quantities likewise constant and are equal, respectively, to the multiplier before \(\lg_n\). In fact, let us take, for example, the curve \(AR\lg_n P\) from equation (12), Fig. 5; differentiate the equation of this curve

Fig. 4.

\[ AR\lg_n P = S, \]

we obtain

\[ AR\frac{dP}{P}=dS, \]

whence

\[ P\frac{dS}{dP}=AR, \]

but \(P\dfrac{dS}{dP}\) is precisely the required subtangent \(bc\), correspondingly \(b'c'\), etc. The indicated circumstance makes it possible at once to detect an error which may creep in when constructing a logarithmic curve by points.

Fig. 5.

Of course, by the subtangent here one must mean the segment cut off by the tangent to the curve at the given point and by the perpendicular dropped from this same point to the coordinate axis toward which the logarithmic curve faces with its convex side, i.e. to the axis which above, in constructing the curves, we took as the axis of ordinates. For example, for the point \(a\), Fig. 5, the subtangent will be \(bc\), and not \(de\). This circumstance should be remembered; without it one may make a mistake by taking the wrong axis.

The second property of these logarithmic curves that is important for us consists in the following: if we put \(const.\) in equations (9), (11), and (12) equal not to 0, but to some positive or negative quantity, then we obtain new curves which will be equidistant from the original curves, but shifted along the axis of ordinates (the axis of the dependent variables) by the amount to which we have equated \(const.\)

For example, Fig. 6 shows the curve \(c_p \lg_n T\), shifted downward by the amount \(c_p \lg_n T_o\), i.e. \(const. = -c_p \lg_n T_o\).

Thus, if in any of the logarithmic curves it is necessary to change \(const.\), then for the new curve no calculations need be made; one simply shifts it point by point to one side or the other. It is only necessary to remember that the shift can be made only along the axis of the dependent variables.

Fig. 6.

Isochore and isobar in the coordinates \(TS\).

Finally, in connection with the preceding properties, it is useful to pay attention to how, in the coordinates \(TS\), the curves \(V = const.\) and \(p = const.\) are expressed, which occur on every diagram of internal-combustion engines—at least on the theoretical diagram, which must always be drawn in the coordinates \(TS\) for comparison with the given actual diagram.

For the isochore \(V = const.\), we have, assuming as above that \(c_v = const.\),

\[ \frac{dQ}{dT} = c_v = const., \]

but by equation (3) \(dQ = TdS\), consequently,

\[ \frac{TdS}{dT}=c_v, \]

whence

\[ dS=c_v\frac{dT}{T}, \]

and after integration

\[ S_v=c_v\, lg_n T+const.; \tag{15} \]

this is the equation of a logarithmic curve, which we have already encountered in equation (9).

In the same way we find, for the isobar \(p=const.\),

\[ S_p=c_p\, lg_n T+const., \]

the equation of a logarithmic curve known to us from equation (12).

Let us now turn to the actual transfer of indicator diagrams into the coordinates \(TS\). At first glance it seems immaterial by means of which logarithmic curves—equation (9), or (11), or (12), i.e. according to the scheme of Fig. 2, or Fig. 3, or Fig. 4—the transfer is to be carried out; in fact this is not so. As will be seen below, equation (11) cannot at all be used if, in accordance with the latest experimental data, the heat capacities \(c_p\) and \(c_v\) are to be regarded not as constant but as increasing with temperature. Of the two remaining equations (9) and (12), in which only one curve depends on the coordinates \(P\) or \(V\), it is more convenient to use equation (12), since in it the logarithmic curve depends on the pressure \(P\), whose scale is easier to reset and which, moreover, generally has to be reset in any case if one wishes to take an average experimental diagram and allow in the calculation for the calibration of the indicator spring, which is rarely quite uniform.

Choice of the equation for the transfer.

Next it should be noted that, in transferring the diagram of a four-stroke engine, it is usually sufficient to take a single working diagram, i.e. one traced by the lines of compression, combustion, and expansion, closing it at the dead point with the vertical line \(ac\), Fig. 7, see p. 10, and discarding the lines of expulsion of the products of combustion \(ab\) and of admission of the fresh charge \(bc\), i.e. the diagram of the charging pump\(^2\). This is done, on the one hand, because the expulsion and suction lines, at the small scale of the spring usually employed in indicating internal-combustion engines, almost merge with one another and with the atmospheric line, and, on the other hand, because

Choice of the indicator diagram.

\(^2\) For a detailed exposition of what part of the recorded diagram should be taken as the indicator diagram of an internal-combustion engine, and why, see: V. L. Maleev, “Calculation of the indicated work of gas engines.” Proceedings of the St. Petersburg Polytechnic Institute, 1905, vol. III, issues 1–2, pp. 159–176.

These lines represent processes with variable quantities of gas (during this time the cylinder is in communication with the atmosphere, respectively through the gas and air passages), and therefore, although when they are transferred into the \(TS\) coordinates the area bounded by them will represent, in heat units, the work of charging, the temperatures \(T\) obtained on this part of the diagram will not represent the actual temperatures of the gases at the corresponding instants, but will be fictitious. The actual temperature \(T\) will be greater than the latter in the ratio of the total weight of the charge \(G\) to the weight \(G'\) of the gases present in the cylinder at the given instant; namely,

\[ T := T' \frac{G}{G'} . \tag{17} \]

Thus the transfer of these strokes into the \(TS\) coordinates is quite useless, and therefore is almost never done.

Fig. 7.

In two-stroke machines the indicator diagram of the working cylinder itself corresponds precisely to the working diagram of four-stroke machines indicated above. True, for the part of this diagram from the beginning of exhaust to the end of admission, the weight of gas present in the cylinder is also a variable quantity, and the temperatures on the heat diagram will not correspond to the actual temperatures; but this must be put up with, on the one hand because making an exact allowance for the weight of the gas during this period is very difficult, if not impossible, and, on the other hand, this period on the diagram is not so very large; finally, the fluctuations of the weight \(G\) are incomparably smaller than during the charging of four-stroke machines. In two-stroke machines \(G_{\min} = 0.6\) to \(0.7G\), whereas in four-stroke machines \(G = 0.25\) to \(0.16G\), decreasing as the compression ratio \(\varepsilon\) increases; in a Diesel engine \(G_{\min} \cong 0.04G\).

Thus, if only the working diagram is transferred—whether of a four-stroke or a two-stroke machine is immaterial—then, as is immediately evident, part of

Graphical construction of thermal diagrams.

one will not have to use the logarithmic curve \(c_p \lg T\) for \(T<273^\circ\); therefore, in order to shorten the work, it need not be calculated, while, to reduce the dimensions of the drawing—which promotes accuracy of construction—this curve may be shifted to the left, on the basis of the property indicated above; see drawing 6, by taking

\[ const.=-c_p \lg 273. \]

The graphical transfer of the diagram now under consideration is to be based on using the characteristic equation for gases

\[ PV=GRT. \tag{14} \]

On the gas constant \(R\).

In this equation the gas constant \(R_1\) of the combustible mixture before ignition is, strictly speaking, not equal to the constant \(R_2\) of the products of combustion. In most cases, owing to the change in the number of gas molecules when the triatomic molecules \(CO_2\) and \(H_2O\) are formed, and, with liquid fuel, also owing to the transition of the liquid into the gaseous state, the constant \(R\) undergoes a certain change. If the coefficient of molecular change \(\beta\) is taken to mean the ratio of the total number of molecules of 1 kg of the products of combustion to their number in the combustible mixture before ignition, then

\[ R_2=\beta R_1. \tag{18} \]

The magnitude \(\beta\) varies depending on the composition of the fuel and the compression ratio \(\varepsilon\), from \(\beta=0.92\text{--}0.93\) for gaseous fuel to \(\beta=1.07\) for liquid fuel. Despite these rather considerable changes in the constant \(R\) before and after combustion, in most practical cases one may take \(\beta=1\), as is done in the following example. It will, however, be indicated below how this change of \(R\) is to be allowed for and expressed on the diagram, although this has to be done only in especially precise investigations.

Thus, in investigating internal-combustion engines by equation (14), the quantities \(P\) and \(V\) will be known from the indicator diagram and from the dimensions—the stroke and the piston diameter—of the engine; the value \(R\) can most simply be determined by analysis of the products of combustion. It remains still to determine \(G\) and \(T\).

In rare cases, chiefly in laboratory investigations or in testing not very large engines, the consumption of air and fuel is determined by direct measurements; then, of course, it is easy to calculate the weight \(G\) of the combustible mixture corresponding to one stroke of the engine. The last unknown quantity \(T\) will then be determined by means of the graphical method described above.

It is somewhat more difficult to proceed in those cases, most common in practice, when \(G\) is unknown. In these cases it is necessary to find

\(T\) for any one point of the indicator diagram, then, from \(P, V, R\) and the \(T\) found, determine \(G\) from equation (14), and thereafter proceed as in the preceding case.

Finding the initial temperature \(T\).

As the initial point for finding \(T\), it is best to take the moment at the beginning of compression.

Let \(G_{\mathrm{mix}}\) denote the weight of the mixture at point \(a\), Fig. 8; \(G_{\mathrm{pr}}\), the weight of the combustion products remaining in the compression chamber at the end of exhaust; and \(G_{\mathrm{in}}\), the weight of the newly inducted air and fuel. Then, obviously,

\[ G_{\mathrm{mix}} = G_{\mathrm{pr}} + G_{\mathrm{in}}. \]

Fig. 8.

Expressing these weights by means of volumes, pressures, and temperatures from the characteristic equations for the corresponding points, and taking the constant \(R\) to be the same \((\beta = 1)\), we obtain

\[ \frac{(V_{\mathrm{in}}+V_c)p_a}{T_a} = \frac{V_c p_b}{T_b} + \lambda_{\mathrm{in}}\frac{V_{\mathrm{in}}p_0}{T_0}, \tag{19} \]

where the corresponding volumes and pressures are shown in Fig. 8; \(T_a\) and \(T_b\) are the temperatures at points \(a\) and \(b\), and \(T_0\) is the temperature of the inducted air and fuel \((T_0 = 273 + t_0^\circ \mathrm{C})\); finally, \(\lambda_{\mathrm{in}}\) is the delivery coefficient during induction, which must be assigned according to experimental data for analogous engines. Some indication as to the magnitude of \(\lambda_{\mathrm{in}}\) can be obtained for a given engine by taking a diagram with a weak spring, Fig. 8, from the ratio \(\dfrac{V_{\mathrm{in}}}{V_c}=\lambda_0\), which may be called the volumetric delivery.

Owing to the influence of the hot walls, \(\lambda_{\mathrm{in}}<\lambda_0\). In general, \(\lambda_0\) varies from 0.93 to 0.80 and even to 0.65, depending on piston speed, the amount of opening of the inlet valves, and their cooling.

From equation (19) we determine \(T_a\) and, substituting \(\dfrac{V}{V_c}=\varepsilon\) and \(\dfrac{V_{\mathrm{in}}}{V_c}=\varepsilon-1\), obtain the required temperature

\[ T_a = T_0 \frac{\varepsilon p_a} {\lambda_{\mathrm{in}}(\varepsilon-1)p_0+\dfrac{T_0}{T_b}p_b}. \tag{20} \]

In this equation (20), the atmospheric pressure \(p_0=\dfrac{B}{735.5}\) kg/cm\(^2\), where

$B$ is the barometer reading during the test of the machine, in mm of mercury column; the pressures $p_a$ and $p_b$ are taken from the diagram, which for this purpose must be taken with a weak spring, giving only the lower part of the diagram, but in a large pressure scale. $T_o$ in equation (20) is measured directly, while the temperature $T_b$ of the combustion products at point $b$ has to be specified; with sufficient accuracy one may take $T_b$ as equal to the temperature of the exhaust gases, measured as close as possible to the cylinder.

Of course, in such investigations it is necessary to know accurately the volume $V_c$ of the compression chamber; for this it is best to measure it directly, by filling it with water or, still better, with oil.

Turning now to the construction of the thermal diagram, it must be noted that its scale must, for the sake of accuracy of construction, be taken sufficiently large; for the temperature one may take, for example, $m_t = 0.05 - 0.5$ mm $= 1^\circ$ C. Since by equation (12) the quantities $m_t$, $m_p$, $m_v$, and $m_R$ are connected, of which $m_p$ and $m_v$ are most often given in advance (the indicator diagram), the scale of the gas constant $m_R$ after choosing $m_t$ is determined by itself; in doing so one must see to it that on the drawing the segment $GRm_R$ does not become small, since in that case the auxiliary lines in finding the temperatures will intersect the axis $OP$ at too acute an angle, and the determination of temperatures, despite a large $m_t$, will be inaccurate. In such a case it is useful to reduce $m_t$ somewhat, which will give a larger $m_R$. In those cases, which are discussed below, when the indicator diagram still has to be redrawn, the scales $m_p$ and $m_v$ should be taken larger: for pressures, approximately $m_p = 10$ mm $= 1$ atm.; the length of the diagram may be taken as $l = 80$ to $120$ mm.

The scale $m_c$, i.e. how many mm $= 1$ heat unit, for the heat capacities $c_p$ and $c_v$, at the same time determining the entropy scale $m_s$ ($m_s = m_c$), should be taken larger, approximately $m_c = 200 - 400$ mm $= 1$ heat unit. In this case, having chosen $m_c$, the logarithmic curves are easiest to construct by referring them to 1 kg; if the weight of the gases performing the cycle is found to be $G$ kg, then the entropy scale will be

\[ m_s = \frac{m_c}{G}. \]

Of course, the heat capacities $c_p$ and $c_v$ are determined from the same gas analysis as the quantity $R$.

Having constructed the logarithmic curves, the indicator diagram is drawn in the lower left quarter, as shown in Fig. 9 (on a separate tab-

...faces), shift the diagram from the origin of coordinates by the amount \(V_c m_v\) and, drawing the atmospheric line at

\[ \frac{B}{735.5}\,m_p \]

above the axis of abscissae.

The actual transfer of the working diagram into the coordinates \(TS\) is carried out point by point, as described above; moreover, if an isobar or an isochore occurs in the indicator diagram, then the accuracy of the construction is easily checked by means of the above-mentioned equations of these lines in the coordinates \(TS\).

When finding points, the subtraction of the corresponding ordinate \(A B \lg_n p\) for acceleration may be done not graphically, as was described, but simply by setting off this quantity with dividers from the curve \(c_p \lg_n T\) on the corresponding isotherms.

Example of transferring a diagram.

In Fig. 9 an example is shown of transferring the diagram of a 25-horsepower Hornsby-Akroyd kerosene engine. All the quantities that served as the basis for the transfer are indicated on the drawing.

In order to evaluate the operation of the engine from which the given diagram was taken, it is useful to draw here as well the diagram of an ideal engine without losses from the influence of the walls and from cooling by the water jacket.

This diagram will be bounded in the coordinates \(PV\), Fig. 9, first, by the adiabatic compression \(a b_o\), whose initial point coincides with the beginning of compression of the actual diagram.

Next comes the isochore \(b_o c_o\) of instantaneous combustion; the final pressure, the point \(c_o\), is most simply found graphically, having first calculated the final combustion temperature \(T_r\) from the compression temperature \(T_c\) and the quantity of heat \(Q\) which the fuel in the cylinder can theoretically develop under the condition of complete combustion, namely

\[ T_r = T_c + \frac{Q}{c_v G_{\mathrm{mix}}}. \tag{21} \]

Having found \(T_r\) and having \(V_c\) and \(G_{\mathrm{mix}} R\), we find graphically the greatest pressure \(p_r\).

Next come the adiabatic expansion \(c_o d_o\) and the exhaust isochore \(d_o a\).

The transfer of the theoretical diagram is greatly simplified by the properties of its component lines: for the compression adiabat in the axes \(TS\), we draw from the initial point \(A\) the vertical straight line \(A B_o\) up to the temperature \(T_c\). The combustion isochore \(B_o C_o\) is constructed by points, of which it is sufficient to take only 4–5, since the curve (logarithmic) proceeds very smoothly, and we bring it up to the temperature \(T_r\). From the point \(C_o\) we again draw the adiabat—the straight line \(C_o D_o\)—down to the temperature \(T_{\mathrm{exh}}\), which is easily found graphically from the point \(d_o\). The exhaust isochore \(D_o D A\) is easy to construct, noting that it will be equidistant from the isochore \(B_o C_o\), shifted to the right by the segment \(D_o D'\).

Comparing the thermal diagram \(AGBCEFDA\) of the given engine with the diagram \(AB_oC_oD_oDA\) of an engine without losses, we discover the following features of this engine: on the compression line \(AB\) the entropy at first increases—the heat is supplied from the cylinder walls, and especially from the walls of the ignition bulb, heated red-hot; but toward the end of the stroke, owing to the increase in the difference between the temperatures of the working mixture and of the water-cooled walls, heat begins to be taken away; the expansion line \(CEFD\) shows, toward the end, near \(E\), a certain increase of entropy; evidently there is afterburning of part of the injected kerosene, which does not have time to burn along the line \(BC\); this delayed afterburning apparently does not even have time to finish by the moment exhaust begins.

Together with general conclusions about the operation of the engine, the heat diagram makes it possible easily to calculate the entire heat balance of the given engine. For this it is only necessary to measure the corresponding areas with a planimeter and, knowing the scale of areas, to convert them into heat units:

1, during compression the walls as a whole supply heat equal to the difference of the areas

\[ a'AGg'a' - g'Gb'g' = 0.17 \text{ heat units}; \]

2, during combustion at dead center, heat was developed equal to the area \(b'BCc'b' = 13.23\) heat units, whereas theoretically there should have been developed the heat \(a'B_oC_oc_o'a' = 19.04\) heat units; the difference of these areas \(= 5.81\) heat units shows the heat lost as a result of incomplete combustion and heat transfer to the walls;

3, the area \(EFf'e'E = 0.40\) heat units represents the heat developed during combustion after dead center—the phenomenon of afterburning;

4, the area \(AGBCEFDA = 3.33\) heat units has been converted into indicated work;

5, the products of combustion have carried away heat equal to the area \(FDAa'f'F = 8.43\) heat units. With regard to the exhaust curve \(FDA\), it must be remembered that, unlike the other lines of the diagram, it represents the change of state of a variable quantity of gas; and therefore, although the area \(FDAa'f'F\) bounded by it represents the heat carried away by the products of combustion, the temperature curve itself is entirely fictitious—the real, true values are possessed only by the initial point \(F\) and the final point \(A\). From this curve it is impossible to determine the temperatures, even using the correction indicated above, equation (17), since it is impossible to compute the quantity of gases \(G'\) present in the cylinder at the given moment;

6, the thermal efficiency \(\eta_t\) of the given engine is

\[ \eta_t = \frac{AGBCFDA}{a'B_oc_oc_o'a'} = 0.175; \]

7, the indicated coefficient \(\eta_i\), showing how close the given machine is to a machine without losses,

\[ \eta_i=\frac{AGBCDA}{AB_oC_oD_oDA}=0.586. \]

Practical use of the method. For practical use the described method can be of significance only if, having calculated and drawn the logarithmic curves for some diagram, they can be used not only for the given case, but also for diagrams taken under entirely different conditions and even from other machines, and, moreover, used with the least expenditure of time, i.e. without redrawing them, but, for example, simply by tracing them onto tracing paper.

As we shall now see, the method just described satisfies these conditions fully, namely if one uses the equation

\[ S=c_p\,lg_n T-c_p\,lg_n 273-AR\,lg_n P, \tag{22} \]

Let us examine, in fact, all the cases when we have diagrams taken under conditions other than the first one, for which the logarithmic curves have been drawn.

Changes of the principal scales. 1, there is a diagram of another length \(l\)—taken with another stroke-reducer,—i.e. the scale of volumes \(m_v\) has changed. This has no influence on the construction of the heat diagram; it is only necessary to lay off the volume of the compression chamber \(V_c\) on the new scale \(m_v\), and to change, according to equation (13), the scale \(m_R\) when laying off the quantity \(GR\) for finding temperatures.

2, if there is a diagram taken with another spring, i.e. with another pressure scale \(m'_p\), then it is simplest to reconstruct its ordinates in the former scale \(m_p\), especially since, ordinarily in calorimetric investigations, it is not the individual diagrams taken during the experiment that are transferred into \(TS\) coordinates, but the mean diagram of the experiment, which is most often constructed from points on the basis of all the diagrams of the given experiment; moreover, when checking the spring scale it often proves to be so non-uniform that in precise investigations the diagram nevertheless has to be reconstructed in some one definite pressure scale. Finally, diagrams of internal-combustion engines usually have to be taken with so stiff a spring (\(m=3.5\) to 1 mm) that, for the sake of accuracy in constructing the \(TS\) diagram, the indicator diagram must be reconstructed; in doing so, the ordinates must, of course, be measured not with a ruler with divisions, but with a vernier caliper with a vernier to 0.1 mm.

Graphical Construction of Thermal Diagrams.

However, in practice a case is possible when, as the “mean transfer diagram,” one may take one of the actual diagrams recorded by the indicator, and the scale of the spring is so uniform and so large that, for its sake, there is no reason to reconstruct the ordinates of the diagram; in such a case the diagram to be transferred into the \(TS\) coordinates may be left unchanged, and the transfer itself carried out by one of the following two methods: first, one may reconstruct the curve \(AR\lg_n p\), shifting its points in accordance with the new scale \(m'_p\); this reconstruction requires no calculations and is performed very quickly. In what follows one need only remember that the scale \(m_R\), owing to the change of \(m_p\), according to equation (13) must also be changed. Secondly, one may proceed as follows: for constructing the temperatures, lay off the value \(GR\) in the new scale \(m_R\), as in the preceding case, and then, for using the curve \(AR\lg_n p\), find for the corresponding points the pressure \(p\), reduced to the initial scale \(m_p\); this can be done either by multiplying the corresponding ordinate by the ratio \(\dfrac{m_p}{m'_p}\),

or graphically, Fig. 10: along the axis \(OT\) lay off the segment \(oa=m_p C\), where \(C\) is an arbitrary multiplier; then lay off \(ab=m'_p C\), draw \(bc \parallel OP\); the required ordinate for the point \(A\), plotted to the scale \(m'_p\), is found by dropping from \(A\) a perpendicular to \(Ad\) onto \(cb\); the intersection of \(ad\) with \(OP\) gives the required ordinate \(oe\), and the line \(ef\) will be the corresponding \(AR\lg_n p\).

Fig. 10.

If \(m'_p<m_p\), point \(B\), Fig. 10, then, laying off \(oa'=m_p C\) and \(a'b'=m'_p C\) and drawing \(b'd' \parallel OP\), we drop a perpendicular to \(Bd'\); the intersection of the continuation of \(a'd'\) with the axis \(OP\) gives the required ordinate \(oe'\). In order not to obscure the further construction, the multiplier \(C\) should be taken sufficiently large (30 to 200), so that the line \(bc\), or respectively \(b'c'\), lies as far as possible below the indicator diagram or, respectively, above the logarithmic curve; the lines \(ad\), or respectively \(a'd'\), are better drawn not in full, but only near the intersection with the axis \(OP\). In all other respects the course of transferring the diagram remains the usual one.

In practice, the first method—the reconstruction of the curve \(AR\lg_n p\)—is perhaps simpler and more convenient, although the second is not complicated either.

Change of \(G\).

3. A change in the weight \(G\) of the working mixture will cause only a corresponding change in the length of the segment \(GR\) in constructing the temperatures, and will also give a change in the scale of heat capacities and, consequently, of the original entropy scale \(m_s\); namely, the new scale \(m_s'\) will be, if \(G'\) is the changed weight of the gases,

\[ m_s'=m_s\frac{G}{G'}. \tag{23} \]

A change in the weight \(G\) of the working gases may occur for the most varied reasons: for example, as a result of a change in the load of the machine, as a result of a change in the resistances in the inlet and exhaust valves and in the pipelines, which, in turn, may occur either from clogging of the pipes or of the muffler, or as a result of a change in the number of revolutions; further, \(G\) may change as a result of a change in the barometric pressure or the pressure in the gas main or in the composition of the gas, or, finally, when \(G\) is computed from equations (14) and (20), as a result of a change in the temperature \(T_a\) at the beginning of compression, which in turn may occur from a change in the temperatures \(T_o\) or \(T_v\) in equation (20).

Change of \(R\) and \(k\).

4. As a result of a change in the operating conditions—a change in combustion or in the composition of the fuel, or even when liquid fuel is used instead of gas, and conversely—the composition of the products of combustion may change; this, in turn, will give a change in the gas constant \(R\) computed from the composition of the products of combustion. In this case two cases are possible: the ratio of the heat capacities of the products of combustion,

\[ \frac{c_p}{c_v}=k, \]

remains the same, and \(k\) changes.

The first case may occur if, when the composition of the products of combustion changes, the percentage content of all the diatomic gases contained in them (chiefly \(O_2\), \(CO\), and \(N_2\)), taken together, has not changed. This follows from the fact that all diatomic gases have one and the same \(k=1.40\), while the polyatomic gases of which the exhaust gases contain chiefly \(CO_2\) and \(H_2O\) in turn also have a common \(k=1.28\).

In this case it is necessary only to take a new segment \(GR'\) for constructing the temperatures and to consider that the entropy scale \(m_s\) has changed, namely,

\[ m_s'=m_s\frac{c_p}{c_p'}=m_s\frac{R}{R'}. \tag{24} \]

In the second case—new \(R'\) and new \(k'\)—one of the logarithmic curves may be left unchanged and it is only to be considered that its scale has changed; the other curve, however, must be reconstructed.

It is more convenient to leave the curve \(c_p \lg_n T\), assuming that its new scale is

\[ m'_s = m_s \frac{c'_p}{c_p}, \]

(but, of course, no longer \(m_s \frac{R}{R'}\)), and to reconstruct the curve \(AR \lg_n p\). The new ordinates of this curve will be

\[ y = AR' \lg_n p \, m'_s . \tag{25} \]

In all other respects one must proceed as usual.

1. If there are diagrams taken from another engine with different dimensions, this is equivalent to having another scale \(m_v'\) and another weight of working gases \(G'\). Thus this case is simply reduced to cases 1 and 3 considered above.

The five cases considered cover all possible cases of variation of the conditions for which the logarithmic curves were originally constructed. As we see, in all cases, when transferring several diagrams, use of the graphical method described gives a particularly considerable saving of time and, moreover, greater assurance in the correct construction of the thermal diagram than does the more laborious analytical calculation of \(T\) and \(S\) for individual points, since checking the graphical construction takes only a few minutes, whereas checking the analytical construction again requires almost as much time as must initially be spent on the construction itself.

It remains only to speak of those cases, encountered almost exclusively in scientific laboratory investigations, when it is desirable to take into account both the change of \(R\) during combustion and also the variability of heat capacities as a function of temperature.

The change of the constant \(R\) can be taken into account very simply by constructing the compression line using \(R_1\), calculated from the composition of the fresh mixture, and the corresponding logarithmic curves, and the expansion line with the aid of \(R_2\), found from the composition of the combustion products, and new logarithmic curves, in which the ordinates of the curve \(AR \lg_n p\) are changed in the ratio \(\frac{R_2}{R_1}\), and the ordinates of the curve \(c_p \lg_n T\) in the ratio \(\frac{c_{p2}}{c_{p1}}\). The combustion line can be found as follows: first construct it for \(R_1\), curve \(ab\), Fig. 11, then for \(R_2\), curve \(cd\); the actual combustion line will lie somewhere between \(ab\) and \(cd\), and the points \(a\) and \(d\) will belong to it; therefore, with sufficient—

with sufficient accuracy one may draw this line by connecting \(a\) and \(d\) with the curve \(ad\), drawn by eye so that it represents a gradual transition from \(ab\) to \(cd\).

Fig. 11.

In view of the fact that the lines \(ab\) and \(cd\) are almost equidistant and lie close to one another, since the effect of the change in the ordinates of the logarithmic curves is comparatively small owing to the fact that \(\beta\) is close to 1 and, moreover, the entropies \(S\) of the corresponding points are equal to the differences of these ordinates, it is possible not to construct the whole line \(cd\), but to confine oneself to constructing the line \(ab\) and only one point \(d\). In this case, as the combustion line one must take only its visible part, for example, in Fig. 9 the line \(BC\); the phenomenon of after-burning on the expansion line is then constructed already according to \(P_2\); the error arising from this is less than the accuracy of the entire construction.

Variable heat capacities. Finally, if in constructing the thermal diagram one abandons the simplifying assumption of constant heat capacities and assumes, in accordance with the latest data, that the heat capacities depend on temperature, and moreover linearly, i.e. adopts

\[ c_p=c_{p0}+bT, \tag{26} \]

\[ c_v=c_{v0}+bT, \tag{27} \]

where \(b\) is a certain constant factor, then this circumstance too is easily taken into account graphically.

Since our equation (12), or (22), was derived for \(c_p=\mathrm{const.}\) and \(c_v=\mathrm{const.}\), it is necessary to return to equation (5). Replacing in equation (5) \(dU\) by expression (6), and substituting \(Pdv\) from equation (17), after differentiating it, we obtain

\[ Pdv+vdP=RdT; \]

then equation (5) takes the form

\[ dQ=c_vdT+ARdT-AvdP; \]

replacing \(v\) from equation (7) by \(\dfrac{RT}{P}\), then \(AR\) by \(c_p-c_v\), and dividing both sides by \(T\), we obtain, as before,

\[ dS=c_p\frac{dT}{T}-AR\frac{dP}{P}. \]

Substituting for \(c_p\) its expression (26), we obtain

\[ dS=c_{po}\frac{dT}{T}+b\,dT-AR\frac{dP}{P}; \]

after integrating this equation we obtain

\[ S=c_{po}\lg_n T+bT-AR\lg_n P+const. \tag{28} \]

In order, as before, to reduce the dimensions of the drawing, one may take the arbitrary constant as \(const=-c_{po}\lg_n 273\); then the required equation is finally obtained in the form

\[ S=c_{po}\lg_n T+bT-AR\lg_n P-5.61\,c_{po}. \tag{29} \]

Since \(AR=c_p-c_v\) for the given gas is a constant quantity even when \(c_p\) and \(c_v\) are variable, the curve \(AR\lg_n P\) remains unchanged; the ordinates of the second curve, however, must be increased, respectively, by the amount \(bt\). This increase is most simply made, again graphically, by laying off the quantities \(c_{po}(\lg_n T-5.61)\) not from the vertical \(OT\), as before, but from the inclined line \(OA\), which forms with the axis \(OT\) the angle \(\alpha\), where \(\tg \alpha=b\), Fig. 12.

Fig. 12.

The procedure itself for constructing the thermal diagram remains entirely unchanged.

For using the indicated variation of the heat capacities, it is useful to indicate their magnitudes, from which the total heat capacity of the working gas can be calculated. According to P. Mollier, for calculating the heat capacities at temperature \(T\) of all diatomic gases \((H_2,\ O_2,\ N_2,\ CO)\) for 1 cubic meter at \(15^\circ\) C and a pressure of \(1\ \mathrm{kg}/\mathrm{cm}^2\),

\[ (C_p)_T=0.268+0.000045\,T, \tag{30} \]

\[ (C_v)_T=0.188+0.000045\,T; \tag{31} \]

in order to obtain the heat capacity referred to \(1\ \mathrm{kg}\) of gas, the indicated expressions must be multiplied by \(\dfrac{24.4}{\mu}\), where \(\mu\) is the molecular weight of the corresponding gas.

Further, the heat capacities of carbon dioxide \(CO_2\), for 1 kg,

\[ (c_p)_T = 0.176 + 0.000106\,T, \tag{32} \]

\[ (c_v)_T = 0.129 + 0.000106\,T; \tag{33} \]

the heat capacities of water vapor \(H_2O\) (superheated), for 1 kg,

\[ (c_p)_T = 0.439 + 0.000154\,T, \tag{34} \]

\[ (c_v)_T = 0.350 + 0.000154\,T. \tag{35} \]

Thus it may be considered proven that the graphical method, when using equations (22) or (29), can be applied to all, without exception, cases of transferring the indicator diagrams of internal-combustion engines into the coordinates \(TS\).

It goes without saying that the graphical procedure described is also applicable in the investigation of other piston machines in which gases are the working fluid: machines operating with hot air, compressors, and blowers, since in them the working conditions are the same and only somewhat simpler, because, owing to the absence of chemical reactions (combustion), the constant \(R\) undergoes no change. In the last two types of machines it is only unnecessary to shift the curve \(c_p \lg_n T\) by the amount \(c_p \lg_n 273^\circ\), since at the end of delivery and at the beginning of suction the fictitious temperature \(T\) (see equation 17) may in them fall to a very low value, \(10^\circ\)—\(5^\circ\) absolute.

Construction of temperature curves.

In conclusion one may also recall that it may sometimes be desirable to follow the thermal changes in an engine by a simplified method, without constructing an entropy diagram, but restricting oneself to the construction only of temperature curves.

On the basis of the characteristic equation (14), the temperature curve can be constructed in a very simple way, externally identical with finding the Derfel characteristics for diagrams of steam engines. As the initial point one may take either the beginning of compression or its end. It is better to take the latter point, since then the temperature scale is larger. Obviously, the graphically corresponding initial volume \(V\) is set equal to \(GR\), and the corresponding pressure \(P = T\).

Before proceeding with the construction, one must find the temperature scale \(m_t\). For this it is first necessary to calculate, by the equation (20) given above, the initial compression temperature \(T_a\). The temperature \(T_c\) needed by us in ...

the end of compression will be found if the characteristic equations for these two points are divided one by the other, in the form

\[ T_c = T_a \frac{p_c V_c}{p_a V_a} = T_a \frac{p_c}{p_a \varepsilon}, \tag{36} \]

where, instead of the pressures \(p_a\) and \(p_c\), one may directly take the corresponding ordinates of the given indicator diagram, while the compression ratio \(\varepsilon\) must be determined beforehand, as in any calorimetric investigation.

Having found \(T_c\) and knowing the pressure scale \(m_p\), we find

\[ m_t=\frac{m_p p_c}{T_c}. \tag{37} \]

In Fig. 13 an example is shown of the construction of temperature curves for the diagram of a Diesel engine.

Example of constructing temperature curves

[Figure labels visible: \(+2000^\circ\) abs.; \(+1500^\circ\); \(+1000^\circ\); \(+500^\circ\); atm.; 35; 25; 15; \(a\); points \(1,2,3,4,5\); \(3'\), \(5'\); “atm. line”; \(T\) during expansion; \(T\) during compression.]

Fig. 13.

For simplicity one may neglect the increase in the weight of the gases owing to the injection of oil; this increase is small, not more than 4%. Likewise one may neglect the change in \(R\) due to combustion. Moreover, in the present case \(\beta < 1\), so that the product \(GR\) in fact changes hardly at all.

For the machine now under consideration, \(\varepsilon=15.4\); \(m_p=1\) mm \(=1\) kg/cm\(^2\). By equation (20) \(T_a=334^\circ\) has been calculated; the initial temperature of the end of compression, by equation (36), is

\[ T_c=334\frac{35}{0.97\cdot 15.4}=782^\circ . \]

Hence the temperature scale, by equation (37), is

\[ m_t=\frac{1.35}{782}\simeq 0.045\ \text{mm}=1^\circ\ \text{C}. \]

The construction itself, as may be seen from Drawing 13, is very simple: first the coordinate axes, the line of zero pressures, and the line of zero volumes are drawn on the diagram; then a vertical \(aa\) is drawn through the end of compression. The temperature at any point \(1\) is found by drawing through it a horizontal line to its intersection with the line \(aa\) at point \(2\); through the origin of coordinates \(O\) and point \(2\) a straight line is drawn until it intersects, at point \(3\), the vertical drawn through point \(1\); the segment \(3\text{–}3'\) is the required temperature. If the straight line \(O\text{–}2\) is continued to its intersection with the vertical drawn through point \(4\) on the expansion line \((p_4=p_1)\), then we obtain the segment \(5\text{–}5'\), the temperature at point \(4\).

From the temperature curves obtained in this way, the following conclusions may be drawn concerning the operation of the given machine: on the compression line, as was to be expected, the increase in temperature, as it approaches the inner dead point, becomes noticeably slower, evidently owing to the increase in heat transfer to the water with the growth of the temperature difference and the decrease in piston speed near the dead point. The temperature curve of combustion and expansion shows, on the one hand, how far the cycle of the given machine is from the original theoretical Diesel cycle with isothermal combustion, and, on the other hand, that combustion continues rather far after the beginning of expansion, i.e. it indicates, in this machine as well, the phenomenon of afterburning.

In concluding this article, we take the liberty of expressing the wish that graphical methods, which in recent times have become so widely used in the most varied fields of technology, may soon obtain rights of citizenship also in the solution of problems of a thermodynamic character, and in particular in the investigation of the operation of internal-combustion engines, since in this field there still remain especially many unresolved questions.

V. L. Michelson. — Construction of thermal diagrams.

Diagram of a four-stroke 25 h.p. Akroyd engine, operating on kerosene in the Laboratory of Heat Engines of the Polytechnic Institute.

Drawing 9.

\[ D = 342.5\ \text{mm}; \]

\[ H = 486\ \text{mm}; \]

\[ T_a = 366^\circ\ \text{abs.}; \]

\[ c_{\text{air}} = 0.05767\ \text{kg}\quad(\text{per 1 stroke}); \]

\[ C_v = 17.92\ \text{kg/hour}\quad(\text{kerosene consumption}); \]

\[ H_u = 10100\ \text{heat units}\quad(\text{calorific value of kerosene}); \]

\[ R_k = 31.9\quad(\text{with }p\text{ in kg/cm}^2,\ R = 0.00319); \]

\[ c_v = 0.278; \]

\[ c_p = 0.242; \]

\[ k = 1.35; \]

\[ n = 200\ \text{rev./min.}; \]

\[ \varepsilon = 2.79; \]

\[ m_p = 3.02\ \text{mm} = 1\ \text{kg/cm}^2; \]

\[ m_v = 1164\ \text{mm} = 1\ \text{atm.}; \]

\[ m_T = 1.0044\ \text{mm} = 1\ \text{heat unit}; \]

\[ m_t = 0.05\ \text{mm} = 1^\circ\text{C}; \]

\[ m_s = 200\ \text{mm} = 1\ \text{m. ed.}; \]

\[ m_d = m_k C_v = 5320\ \text{mm} = 1\ \text{m. ed.}; \]

\[ m_q = 1\ \text{m. ed.} = 260\ \text{mm}^2; \]