A. V. Kvasnikov.

Heat Transfer in a Polytropic Process.

A polytropic process in thermodynamics may be defined as a process one of whose mathematical expressions is identical with the equation of a polytrope, a curve that is mathematically strictly defined. Such an expression is the formal mark of a polytropic process. Using the characteristic equation and the expression of the first law of thermodynamics, one may pass from the equation of the polytrope to the establishment of the law of heat exchange between the working body and the surrounding medium, etc.—to the establishment, for the process, of signs of a thermodynamic character. The latter signs are especially important for the understanding of the process; they speak directly of the character of the conditions of the mutual transformation of thermal and mechanical energy. Conversely, proceeding from these signs, by the same route one may arrive at a relation between two quantities determining the state of the body, which will be expressed by the equation of a polytrope. To use the expression of the relation between the pressure \(p\) and the specific volume \(v\) presents great practical convenience, and in a polytropic process this relation has the form \(pv^n=\mathrm{const.}\)

In the chain that connects the equation of the polytrope with the expression of the law of heat exchange stand the characteristic equation and the first law of thermodynamics; and it is clear that, if these intermediate links are changed, the former character of heat exchange taken as a premise will give another relation \(p=f(v)\), and conversely, the equation \(pv^n=\mathrm{const.}\) after analysis will encounter the assumption of other, not former, premises. Such changes in the expression of the first law occurred when the dependence of heat capacity on temperature was introduced. Processes formerly identical, characterized by a definite law of heat exchange or by the corresponding equation \(pv^n=\mathrm{const.}\), became different, which at first introduced confusion. The exponent \(n\), assigned to the assumed process on the basis of experiment and assumptions about the direction of heat exchange, in fact determined heat exchange directed exactly oppositely, or agreeing with the assumptions only in part of the process. With constant heat capacity it had been quite easy to draw conclusions about a change in the character of heat exchange from a change in the exponent \(n\); this is sometimes still done even now, but the results of such interpretations, as before, are already incorrect. What deviations there may be in the interpretations and results is seen from the following.

Polytrope for \(C_v=\mathrm{const.}\)

With constant heat capacity, the concept of a polytropic process, based on premises of a thermal character, has a quite definite meaning. In this case a polytropic process is called such a change of the state of a body in which the change of its temperature \(T\) is directly proportional to the increment (or withdrawal) of heat \(Q\). As its condition one may write

\[ dQ = C\,dT, \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (1), \]

where the constant quantity \(C\) may be called the “heat capacity” in the given process. The polytrope may also be defined as a curve characterizing a process in which the change of work \(L\) is directly proportional to the change of heat \(Q\), i.e.

\[ dQ = z A p\,dv, \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2). \]

This will be seen if, according to the first law of thermodynamics, we write

\[ dQ = C\,dT = C_v\,dT + A p\,dv, \]

whence

\[ A p\,dv = (C - C_v)\,dT, \]

and

\[ \frac{C}{C - C_v}\,A p\,dv = C\,dT = dQ, \]

i.e.

\[ \alpha = \frac{C}{C - C_v}. \tag{3} \]

Hence it is clear that, in a polytropic change, the ratio of the increment of the internal energy \(U\) to the increment of the work \(L\) is a constant quantity. Introducing the notation \(C = \beta C_v\), we obtain:

\[ dQ = \beta C_v\,dT = \alpha A p\,dv,\quad \text{or}\quad \beta\,dU = \alpha A\,dL, \]

whence

\[ \frac{A\,dL}{dU} = \frac{\beta}{\alpha}, \]

but from (3)

\[ \alpha = \frac{\beta}{\beta - 1}, \]

after which

\[ \frac{A\,dL}{dU} = \beta - 1,\quad \text{also}\quad \frac{A L}{U_2 - U_1} = \beta - 1. \tag{4} \]

To obtain the equation of the polytrope in the coordinates \(pv\), we use the equation written above:

\[ A p\,dv = (C - C_v)\,dT. \]

Substituting for \(dT\) its value \(\dfrac{d(pv)}{R}\) and dividing both sides of the equation by the product \(pv\), we obtain:

\[ A R\,\frac{dv}{v} = (C - C_v)\,\frac{d(pv)}{pv}, \]

and hence

\[ (C_p - C_v)\ln v = (C - C_v)\ln(pv) + \text{const.}, \]

which gives

\[ p v^{\frac{C - C_p}{C - C_v}} = \text{const.} = p v^n, \tag{5} \]

where

\[ n = \frac{C - C_p}{C - C_v}. \tag{6} \]

Conversely, for \(C\) we obtain from this

\[ C = C_v\,\frac{n - k}{n - 1}, \tag{7} \]

and, relating expressions (3) and (6), we determine \(n\) through \(\alpha\):

\[ n = \alpha - k(\alpha - 1), \tag{8} \]

whence

\[ \alpha = \frac{n - k}{1 - k}. \tag{9} \]

Using the characteristic equation \(pv = RT\), we usually find:

\[ \left[\frac{v_1}{v_2}\right]^{n-1} = \frac{T_2}{T_1} = \left[\frac{p_2}{p_1}\right]^{\frac{n-1}{n}}. \tag{10} \]

The magnitude of the work expressed by the shaded area (Fig. 1) may be determined from equation (5), or, after substituting expression (7) into the equation

\[ A\,dL=(C-C_v)\,dT, \]

whence we obtain

\[ L=\frac{C_v}{A}\frac{1-k}{n-1}(T_2-T_1); \]

or

\[ L=\frac{1}{n-1}(p_1v_1-p_2v_2). \tag{11} \]

The most customary expressions for the magnitude of the work are:

\[ L=\frac{p_1V_1}{n-1}\left[1-\left(\frac{p_2}{p_1}\right)^{\frac{n-1}{n}}\right] \quad\text{and}\quad L=\frac{p_1V_1}{n-1}\left[1-\left(\frac{V_1}{V_2}\right)^{n-1}\right]. \tag{12} \]

They are easily obtained from (11) and (10).

For a given curve \(pv^n=\mathrm{const.}\), it is necessary, besides \(L\), also to find \(Q\), the quantity of heat which during the process the body received from without, or gave up. According to (2),

\[ Q=\alpha AL. \]

Substituting the value of \(\alpha\) from (9), we obtain

\[ Q=A\frac{n-k}{1-k}L. \tag{13} \]

To determine the magnitude \(Q\), one may also make use of Fig. 1. By (13) the shaded area gives, on a certain scale, this magnitude. This same area also gives the change in internal energy, since, according to (4),

\[ U_2-U_1=\frac{A}{\beta-1}L=A(\alpha-1)L; \]

\[ U_2-U_1=A\frac{n-1}{1-k}L. \tag{14} \]

The temperatures may be found in the same diagram by the well-known graphical construction for solving the equation \(pv=RT\). From the magnitude of the subtangent it is possible to find the polytropic exponent. Thus all the principal questions connected with the study of the process can be solved graphically in the coordinates \(pv\). In considering complex processes, of which a polytropic process may form a part, the coordinates \(TS\) are now very often used. With a grid drawn in these coordinates, consisting of the lines \(p=\mathrm{const.}\) and \(v=\mathrm{const.}\), the fixed position of the polytrope also makes it possible to solve questions concerning changes in \(p\), \(v\), \(T\), \(U\), to find the work \(L\), the quantity of heat \(Q\), and the exponent \(n\). In \(TS\) coordinates, the polytrope is a logarithmic curve. If

\[ dS=\frac{dQ}{T} \quad\text{and}\quad dQ=C\,dT, \]

then

\[ dS=C\frac{dT}{T}\quad\text{and}\quad S=C\ln T+\mathrm{const.} \tag{15} \]

In Fig. 2 are shown all the quantities which may be of interest in studying the polytrope in the coordinates \(TS\)\(^1\). The question of heat exchange during the process is one of the most important. How the character of heat exchange changes when the polytropic exponent \(n\) changes may be concluded from the expression for \(C\):

\[ C=C_v\frac{n-k}{n-1}; \]

whence

\[ Cn-C-C_vn+C_vk=0. \tag{16} \]

\(^1\) See in detail Ostertag, Die Entropietafel für Luft.

This will be the equation of a hyperbola in the coordinates \(C n\); its center is located at the intersection of the straight lines

\[ n=1 \quad \text{and} \quad C=C_v . \]

For air \((k=1.4;\ C_v=0.175.)\) the hyperbola is shown in Fig. 3.

Its upper branch is suitable for \(n>1\). For \(1<n<k\), \(\dfrac{dQ}{dT}\) is negative, i.e., when heat is supplied the temperature falls, and conversely; moreover, insignificant changes in the quantity \(n\) strongly affect the magnitude of \(\dfrac{dQ}{dT}\). For \(n>k\), changes in \(\dfrac{dQ}{dT}\) are considerably smaller and, for \(n>5\), practically scarcely noticeable. Within these limits, for \(n>k\) and up to infinity, the signs of \(dQ\) and \(dT\) are the same.

The second, lower, branch of the hyperbola indicates that for \(n<1\), \(\dfrac{dQ}{dT}\) is always positive, i.e., the supply of heat causes an increase of temperature, and conversely. The heat exchange changes most sharply, in quantitative terms, when \(n\) varies within the range \(0\text{–}1\). Changes in the exponent \(n\) are often used to judge changes in the character of heat exchange, but this judgment is of a purely qualitative nature, \(^{1)}\) and is far from sufficient for an understanding of heat exchange. This is clear from Fig. 3 as well, and is seen directly from the magnitude of \(\dfrac{dC}{dn}\). From equation (16)

\[ \frac{dC}{dn}=\frac{C_v-C}{n-1}, \quad \text{but} \quad C=C_v\frac{n-k}{n-1}; \]

therefore

\[ \frac{dC}{dn}=C_v\frac{k-1}{(n-1)^2}, \quad \text{or} \quad \frac{dC}{dn}=\frac{\text{const.}}{(n-1)^2}. \tag{17} \]

Hence it is clear that a change of the exponent \(n\) by one and the same amount, when its absolute value is large, will give insignificant changes in the magnitude \(C\), and when it is small—large ones; the most significant fluctuations in the magnitude \(C\) are obtained when \(n\) is near unity.

Polytrope for \(C_v \ne \mathrm{const.}\)

Retaining the first definition of the polytrope, we have:

\[ dQ=c\,dT. \]

For \(C_v=a_v+bT\), the expression of the first law is obtained in the form:

\[ c\,dT=a_v\,dT+bT\,dT+A p\,dv. \]

Dividing both sides of the equation by \(pv\), or \(RT\), and simplifying, we obtain:

\[ (a_v-c)\frac{dT}{T}+b\,dT+AR\frac{dv}{v}=0, \]

and after integration—

\[ (a_v-c)\ln T+bT+AR\ln v=\mathrm{const.} \tag{18} \]

Putting

\[ \frac{b}{a_v-c}\,T=\ln G, \quad \text{i.e.} \quad G=e^{\frac{b}{a_v-c}T}, \]

instead of (18) we shall have:

\[ \ln pv+\frac{AR}{a_v-c}\ln v+\ln G=\mathrm{const.} \]

and hence

\[ pv^{\frac{c-a_p}{c-a_v}} e^{\frac{b}{a_v-c}\cdot \frac{pv}{R}}=\mathrm{const.} \]

\(^{1)}\) For example, V. Leinweber, “Diagramm-Charakteristiken,” Z. d. V. d. I., 1913, p. 534, 1988.

If, analogously to equality (6), we denote

\[ \frac{c-a_p}{c-a_v}=n, \]

then for the equation of the polytrope we obtain

\[ p v^n e^{\frac{b}{a_v-c}T}=\text{const}. \tag{19} \]

In contrast to equation (5), here in the variable part there is the variable factor

\[ e^{\frac{b}{a_v-c}T}, \]

which, as we shall see below, at high temperatures introduces significant changes into the course of the curve. Let us compare, for the same initial state, \(n\), and final temperature, the quantities \(L\), \(\Delta U\), and \(Q\) in processes (5) and (19). In both cases:

\[ Q=c(T_2-T_1)=a_v\frac{n-k}{n-1}(T_2-T_1), \tag{20} \]

assuming that \(c_v=a_v\). Comparing \(\Delta U\), we obtain:

\[ \Delta U_{\mathrm I}=a_v(T_2-T_1) \quad\text{and}\quad \Delta U_{\mathrm{II}}=a_v(T_2-T_1)+\frac{b}{2}(T_2^2-T_1^2). \]

That is, the change in the absolute magnitude of the internal energy in the second case will be greater, and correspondingly the work, for a given temperature difference, will be smaller by the same amount. In Fig. 4 an approximate relative position of the isochores in the \(TS\) coordinates is given for both cases. The shaded area gives the difference \(\Delta U_{\mathrm{II}}-\Delta U_{\mathrm I}=AL_{\mathrm I}-AL_{\mathrm{II}}\). The magnitude of the work \(AL_{\mathrm{II}}\) is determined by the difference \(Q-\Delta U_{\mathrm{II}}\).

\[ AL_{\mathrm{II}} = a_v\frac{n-k}{n-1}(T_2-T_1) -a_v(T_2-T_1) -\frac{b}{2}(T_2^2-T_1^2). \]

\[ AL_{\mathrm{II}} = a_v\frac{1-k}{n-1}(T_2-T_1) -\frac{b}{2}(T_2^2-T_1^2). \tag{21} \]

The ratio between \(AL\) and \(Q\) will therefore have the form:

\[ \frac{AL_{\mathrm{II}}}{Q} = \frac{1-k}{n-k} - \frac{b}{a_v}\frac{n-1}{n-k}\frac{T_2+T_1}{2}, \]

or

\[ AL_{\mathrm{II}}=(\gamma+\delta T_m)Q, \tag{22} \]

where \(T_m\) is the mean temperature over the process.

Let us now find the dependence \(p=f(v)\) for a process in which condition (2) is satisfied, i.e., for which

\[ dQ=\alpha A p\,dv. \]

According to the first law,

\[ \alpha A p\,dv=a_v\,dT+bT\,dT+A p\,dv; \]

dividing by \(pv\), or \(RT\), we obtain

\[ a_v\frac{dT}{T}+b\,dT+AR(1-\alpha)\frac{dv}{v}=0, \]

and after integration:

\[ \ln T+\frac{b}{a_v}T+\frac{AR}{a_v}(1-\alpha)\ln v=\text{const}. \tag{23} \]

But

\[ \frac{AR}{a_v}(1-\alpha)=\frac{a_p-a_v}{a_v}(1-\alpha)=(k-1)(1-\alpha); \]

and after eliminating \(T\) from equation (23) we finally obtain

\[ p v^{\,a-k(\alpha-1)} e^{\frac{b}{a_v R_v}pv}=\text{const}. \tag{24} \]

Introducing a notation analogous to (8), we may write:

\[ p v^n e^{\frac{b}{a_v}T}=\text{const}. \tag{25} \]

Equations (19) and (25) are different. The assumptions on which their derivation rests are not identical, as is generally the case with constant heat capacity. In the latter case (25), just as with constant heat capacity, simple relations exist between the quantities \(\Delta U\), \(AL\), and \(Q\).

Thus

\[ Q=\alpha AL=\frac{n-k}{1-k}AL. \tag{26} \]

From

\[ dQ=\alpha A p\,dv=dU+A p\,dv \]

we obtain

\[ dU=A(\alpha-1)\,dL,\quad \text{i.e.}\quad dU=\frac{n-1}{1-k}A\,dL. \tag{27} \]

Also

\[ dQ=dU+\frac{dQ}{\alpha},\quad \text{i.e.}\quad dU=\frac{\alpha-1}{\alpha}\,dQ, \]

and therefore

\[ dU=\frac{n-1}{n-k}\,dQ. \tag{28} \]

The quantities \(Q\) and \(AL\) are easily computed when the temperature difference is known, namely

\[ Q=\frac{n-k}{n-1}\left[a_v(T_2-T_1)+\frac{b}{2}(T_2^2-T_1^2)\right] . \tag{29} \]

and

\[ AL=\frac{1-k}{n-1}\left[a_v(T_2-T_1)+\frac{b}{2}(T_2^2-T_1^2)\right]. \tag{30} \]

Given the work curve, it can be computed from the expression that is readily obtained from (30) after replacing \(T\) by its value \(\dfrac{pV}{R}\),

\[ L=\frac{p_1V_1-p_2V_2}{n-1} \left[1+\frac{b}{2a_v}(p_1V_1+p_2V_2)\right]. \tag{31} \]

From a comparison of the three cases considered, it is clear that equations (5) and (25) characterize processes similar in their internal content. Namely, they are expressions for such processes in which the change in the quantity of heat \(Q\) is always divided in one and the same constant ratio between changes in the amount of internal energy \(U\) and the work \(AL\).

Let us now consider the relative position of the curves discussed in the coordinates \(TS\). For cases (5) and (19) we shall have:

\[ S=c\ln T+\mathrm{const}. \]

For the latter case (25), according to equality (28):

\[ dQ=\frac{n-k}{n-1}\,dU=\frac{n-k}{n-1}(a_v+bT)\,dT; \]

whence

\[ dS=\frac{n-k}{n-1}\left(a_v\frac{dT}{T}+b\,dT\right) \]

and therefore

\[ S=\frac{n-k}{n-1}\left(a_v\ln T+bT\right)+\mathrm{const}. \tag{32} \]

With variable heat capacity, the coordinates \(ST\), as is evident, give simpler equations than the coordinates \(pV\). Therefore, when using the \(TS\) and \(pV\) diagrams jointly, it is better to construct the \(TS\) curve and then transfer it to \(pV\). The transfer itself can be carried out as was done in Fig. 5. The coordinate axes are arranged in such a way that the axes \(S\) and \(v\) are on the same—

straight line, and the axes \(p\) and \(T\) coincide. In the coordinates \(TS\), besides the polytrope, we have some curve \(v=\mathrm{const.}\) (in the drawing, passing through point \(A\)). Along the \(v\)-axis we lay off \(v_a\), \(a\), and also \(R\), after which, by the construction indicated in the drawing, we find \(p_a\), i.e. point \(A\). Taking \(v_a\) as unity, we construct the auxiliary curve \(A R \ln v\) along the \(v\)-axis. In transferring point \(C\) we note that

\[ DC=S_c-S_d=AR\ln \frac{v_c}{v_d}=AR\ln v_c, \]

since \(v_d=v_a=1\).

We transfer the segment \(DC\) onto the curve \(AR\ln v\) and find the corresponding \(v\), and then, knowing \(v_c\), \(T_c\), and \(R\), graphically find \(p_c\), just as for point \(A\). In Fig. 6 the curves (5), (19), and (25) are constructed in the coordinates \(TS\) and transferred by the indicated method into the coordinates \(pv\). The initial state is assumed identical. For constant heat capacity we have curves I and \(v_1\). For the case (19), curves II and \(v_{\mathrm{II}}\), and, finally, the last case is given by curves III and \(v_{\mathrm{III}}\). The example given in the drawing is for air, with the following values adopted:

\[ n=1.6,\quad k=29.27,\quad c_v=0.162+0.0000366\,T,\quad p_a=40\ \mathrm{atm.},\quad T_a=2100^\circ\mathrm{abs.} \]

Comparing cases 1 and 3, we find that, at the same final temperature, more heat is lost in the latter case, and the work obtained must be greater than the following, since

\[ \frac{L_{\mathrm{III}}}{L_i} = 1+\frac{b}{a_v}\frac{T_2+T_1}{2} = 1+\frac{b}{a_v}T_m, \]

where \(T_m\) is the mean temperature over the process. (In the present example \(L_{\mathrm{III}}=1.35\,L_i\).) This ratio does not depend on the exponent \(n\). Such a dependence is found for the ratio

\[ \frac{L_{\mathrm{II}}}{L_i} = 1-\frac{b}{a_v}\cdot\frac{n-1}{1-k}T_m. \]

For \(n>1\), \(L_{\mathrm{II}}>L_i\), and conversely (in the example \(L_{\mathrm{II}}=1.53\,L_i\)).

The difference in the magnitude of the work is also considerable for the same final volume. In the present example

\[ L_{\mathrm{II}}:L_{\mathrm{III}}:L_i=116:109:100. \]

From equations (19) and (25),

\[ pv^n e^{-\frac{b}{a_v}\frac{n-1}{k-1}T}=\mathrm{const.} \quad\text{and}\quad pv^n e^{-\frac{b}{a_v}T}=\mathrm{const.}, \]

it is not difficult to obtain equations for the principal simplest processes.

For an isochore we have \(v=\mathrm{const.}\), i.e. \(n=\pm\infty\) for the case according to equation (25). This condition gives

\[ \alpha-k(\alpha-1)=\pm\infty,\quad\text{i.e.}\quad \alpha=\pm\infty. \]

In order that equation (19) give \(v=\mathrm{const.}\), it is necessary already to satisfy two conditions:

\[ n=\pm\infty,\quad \text{and}\quad b=0. \]

For \(n=\pm\infty\) we have

\[ \frac{a_v-c}{a_p-c}=0, \]

whence

\[ c=a_v. \]

Thus, the condition \(dQ=c\,dT\) and the given dependence of heat capacity on temperature \((b\ne 0)\) cannot give a process at constant volume.

For the isobar \(p=\mathrm{const.}\) in the case (19) we have

\[ c=a_p,\quad b=0; \]

in the case (25)—

\[ \alpha-k(\alpha-1)=0,\quad\text{i.e.}\quad \alpha=\frac{k}{k-1}, \]

and the additional condition \(b=0\), i.e., with variable heat capacity one cannot obtain a process at constant pressure under the conditions

\[ dQ=c\,dT,\quad \text{or}\quad dQ=\alpha A p\,dv. \]

Having the process \(p=\mathrm{const.}\) and a variable heat capacity, we obtain between \(Q\) and \(L\) no longer a linear dependence, which is easily seen if the conditions \(p=\mathrm{const.}\) and \(c_v=a_v+bT\) are introduced into the equation

\[ dQ=a_v\,dT+bT\,dT+Ap\,dv. \]

Using the characteristic equation, we first eliminate \(T\):

\[ dQ=\frac{a_v}{R}p\,dv+\frac{b}{R^2}pv\cdot p\,dv+Ap\,dv =\frac{ap}{R}\,dL+\frac{b}{R^2}pv\,dL, \]

but

\[ L=p(v-v_0),\quad pv=L+pv_0. \]

Substitute this value of \(pv\) into the expression for \(dQ\):

\[ dQ=\frac{ap}{R}\,dL+\frac{b}{R^2}L\,dL+\frac{b}{R^2}pv_0\,dL =\frac{apR+bpv_0}{R^2}\,dL+\frac{b}{R^2}L\,dL, \]

and hence

\[ Q=\frac{ap+bT_0}{R}L+\frac{b}{2R^2}L^2. \tag{33} \]

The constant is equal to zero, since \(Q\) is counted only during the process, i.e. \(Q=0\) when \(L=0\). In Fig. 7 a part of the parabola \(Q=f(L)\) (33) is plotted, with \(L\) reckoned along the horizontal axis.

Passing to the condition \(T=\mathrm{const.}\), we find from \(dQ=c\,dT\), \(c=\pm\infty\), and from equality (27), \(n=1\) and \(\alpha=1\), which gives for cases (19) and (25) the equation of the isotherm

\[ pv=\mathrm{const.} \]

For the adiabat the fundamental condition is \(dQ=0\); i.e., for cases (19) and (25) there will successively be

\[ c=0\quad\text{and}\quad \alpha=0, \]

after which we obtain the equation of the adiabat with variable heat capacity:

\[ pv^k e^{\frac{b}{a_v}T}=\mathrm{const.} \tag{34} \]

At high temperatures curve (34) differs appreciably from the curve \(pv^k=\mathrm{const.}\) This is of great importance in estimating theoretical cycles for internal-combustion engines. The use of equation (34) instead of the usual equation of the adiabat introduces very substantial corrections both in the interpretation of theoretical diagrams and in the magnitude of the thermal coefficient of useful action¹. In Fig. 8, as an example, the curves \(pv^k=\mathrm{const.}\) and

\[ pv^k e^{\frac{b}{a_v}T}=\mathrm{const.} \]

are given (air, \(p_a=40\) atm., \(T_a=2100^\circ\) abs.). On expansion to the same volume \(v_b\), we find that in the second case there is an excess of area of approximately \(12\%\). The temperature at the end of expansion is, for \(c_v=\mathrm{const.}\), \(980^\circ\) abs., and for nonconstant heat capacity, \(1200^\circ\) abs. The differences are quite considerable.

\[ pv^n=\mathrm{const.}\quad \text{with } c_v\ne \mathrm{const.} \]

For this case the following equations are valid:

\[ dQ=c_v\,dT+Ap\,dv,\quad pv^n=\mathrm{const.},\quad pv=RT. \]

Differentiating the last two, we eliminate \(vdp\) from them and, having determined \(pdv\), substitute its value into the first:

\[ vdp+npdv=0,\quad pdv+vdp=R\,dT, \]

\[ pdv=\frac{R}{1-n}\,dT, \]

\[ dQ=\left[c_v+\frac{AR}{1-n}\right]dT=c\,dT. \tag{35} \]

The quantity \(C\) is variable, namely:

\[ c=a_v+\frac{AR}{1-n}+bT=g+bT, \tag{36} \]

where

\[ g=a_v\frac{n-k}{n-1}; \tag{37} \]

whence

\[ n=\frac{g-a_p}{g-a_v}, \tag{38} \]

i.e. the equation of the polytrope will be:

\[ pv^{\frac{g-a_p}{g-a_v}}=\mathrm{const}. \tag{39} \]

Expressions (37), (38), and (39), as is evident, fully coincide with the corresponding ones for constant heat capacity, if one puts, as was done earlier, \(a_v=c_v\) \((c_v=\mathrm{const}.)\). Here too, as for \(c_v=\mathrm{const.}\), we easily obtain the simplest special cases. Thus, for \(g=a_p\) we obtain an isobar, for \(g=a_v\)—an isochore, and for \(g=\infty\)—an isotherm. For \(g=0\) we obtain the curve \(pv^k=\mathrm{const.}\), but this will no longer be an adiabat, as it should be for constant heat capacity. Since \(b\) is usually a positive quantity, this curve, for expansion, indicates that the body loses a certain quantity of heat, which, according to expression (35),

\[ dQ=\left[a_v\frac{n-k}{n-1}+bT\right]\,dT \tag{35-a} \]

in our case gives:

\[ Q=\frac{b}{2}(T_2^2-T_1^2). \]

It is easy to see that

\[ \frac{dQ}{T}=dS=b\,dT, \]

and

\[ S=bT+\mathrm{const.} \]

i.e. in the coordinates \(TS\) the curve \(pv^k=\mathrm{const.}\) will be an inclined straight line (see Figs. 14 and 15). The ratio of the quantity of heat in this process to the work produced is rather large.

\[ AL=Q-\Delta U=\frac{b}{2}(T_2^2-T_1^2)-a_v(T_2-T_1)-\frac{b}{2}(T_2^2-T_1^2)= \]

\[ =-a_v(T_2-T_1); \]

i.e.

\[ \frac{Q}{AL}=-\frac{b}{2a_v}(T_2+T_1),\quad \text{or}\quad \frac{Q}{AL}=-\frac{b}{a_v}T_m, \]

where \(T_m\) is the mean of the limiting absolute temperatures. Having, for example, expanding combustion products with \(T_1=2200^\circ\) and \(T_2=800^\circ\), we obtain

\[ \frac{Q}{AL}=\frac{0{,}0000533}{0{,}162}\,1500=0{,}494. \]

In the general case the heat exchanged by the body during the process will be

\[ Q=g(T_2-T_1)+\frac{b}{2}(T_2^2-T_1^2). \tag{40} \]

From this expression it is evident that heat in this process, at one and the same exponent \(n\), may be supplied to the body and removed from it. In Fig. 9, for finding \(Q\), there are two curves \(gT\) and \(\dfrac{b}{2}T^2\). The straight lines \(gT\), for positive \(g\), pass through the left quadrant, and for negative \(g\)—through the right.

This makes it possible automatically to carry out the algebraic addition of the quantities \(gT\) and \(\dfrac{b}{2}T^2\). Let us call

\[ W = gT + \frac{b}{2}T^2, \tag{41} \]

then, for positive \(g\), \(W\) always increases as the temperature increases, i.e. the heat in the imaginary polytropic process from absolute zero is always received by the body.

Negative \(g\) gives at first, as the temperature increases, a negative \(W\), and later a positive one. The intersection of the lines \(gT\) and \(\dfrac{b}{2}T^2\) determines the temperature \(T_e\), which is obtained as the final temperature in a process with \(W=0\). We find its value from the condition

\[ g + \frac{b}{2}T_e = 0,\quad \text{whence } T_e = -\frac{2g}{b}. \tag{42} \]

To find the value of \(Q\) we shall have

\[ Q = W_2 - W_1. \]

In Fig. 9, in this way, \(Q\) has been found for the particular case \(Q\) (products of combustion, \(C_v = 0.162 + 0.0000533\,T\), \(n = 1.35\)). If, during the process, \(Q\) is equal to zero, then

\[ g(T_2 - T_1) + \frac{b}{2}(T_2^2 - T_1^2) = 0, \]

whence

\[ \frac{T_1 + T_2}{2} = T_m = -\frac{g}{b}, \tag{43} \]

i.e. \(Q\) can be equal to zero in such a polytropic process when the mean of the extreme temperatures is equal to \(-\dfrac{g}{b}\) (twice smaller than \(T_e\)), or also when it is equal to the temperature corresponding to the zero heat capacity (apparent). Indeed, if \(C = 0\), then

\[ g + bT = 0 \quad \text{and} \quad T = -\frac{g}{b} = T_m. \]

The point lying on the polytrope that corresponds to this temperature divides the curve into sections of heat removal and heat supply. For each exponent \(n\), the temperature \(T_m\) will be its own, just as for each gas composition. We find the relation between \(T_m\) and \(n\) by substituting, instead of \(g\), its value from (37) into expression (43). Then

\[ T_m = \frac{av}{b}\,\frac{k-n}{\,n-1\,}. \tag{44} \]

In Fig. 10, for the particular case (products of combustion), the curves \(gn\) and \(nT_m\) are given. From it one sees that, practically, the temperature \(T_m\) can be observed for polytropes with exponents smaller than \(k\) and not far from it. In the example cited it is seen that at \(n = 1.25\), \(T_m\) already obtains the value \(1825^\circ\) abs. At \(n = 1.35\), from Fig. 9, for the same products of combustion we have \(T_m = 550^\circ\) abs.

Comparing the quantities of heat calculated for processes according to \(pv^n = \text{const.}\) with constant and variable heat capacities, we see that ignoring the dependence of heat capacity on temperature is the same as ignoring the quantity \(\dfrac{b}{2}T^2\) calories. The ratio of this quantity to that usually calculated at \(C_v = \text{const.}\) will be (for simplicity, we take the imaginary process from \(T = 0\)):

\[ \frac{b}{2}T^2 : gT = \frac{b}{2av}\,\frac{n-1}{\,n-k\,}\,T = r. \]

In Fig. 10 the values of $\gamma$ are plotted with a dotted line for $T = 2000^\circ$ abs. The unit in which $\gamma$ is measured in the drawing is indicated by the dotted vertical. Recognizing that, when the correction term

\[ \frac{b}{2}T^2 \]

amounts to $6\%$ or more, it can no longer be neglected, we obtain for the limiting values of $g$:

\[ \frac{bT^2}{2}=\pm 0.06\,gT,\qquad g=\pm\frac{bT}{2\cdot 0.06}, \]

and for our example $g=\pm 0.888$. Determining $n$, we obtain: $n_1=0.91$ and $n_2=1.06$; that is, at $T_{\max}=2000^\circ$ in polytropic processes with combustion products of the composition taken, one may use the equation $pv^n=\mathrm{const.}$, with arguments based on $C_v=\mathrm{const.}$, only within the limits $n=0.91\text{--}1.06$. These limits are marked on the drawing by two lines (dot-and-dash). They are very small and in practice have almost no significance, since those average values of $n$ which it makes sense to use in the projected, assumed cycles usually do not lie within these limits.

In Fig. 11, with the curve

\[ \frac{b}{2}T^2 \]

the position of the straight lines $gT$ is shown for various exponents $n$, and certain values of $W$ with their signs are also indicated. In the $TQ$ diagram, for the curve $pv^n=\mathrm{const.}$, the quantities $Q$ and $AL$ are easily found in the form of line segments. For the cases $n>k$ and $n<k$ this is done in Figs. 12 and 13. When the temperature is raised from $T_1$ to $T_2$, the internal energy increases by an amount expressed by the segment $ac$ (also $a_1c_1$ for $n$ considerably less than $k$, Fig. 13). The heat that has participated in the exchange is found, as before, in the form of the segment $ad$ ($a_1d_1$ is negative, i.e. heat is lost). The difference $Q-\Delta U=AL$

\[ ad-ac=cd,\quad (\text{also } c_1d_1), \]

is expressed by a segment which, on a certain scale, gives the required work of compression.

The relation between $Q$ and the work performed may be found as follows:

\[ A\,dL=dQ-dU=(g-a_v)\,dT = a_v\frac{1-k}{\,n-1\,}\,dT,\qquad \ldots \qquad (45) \]

and

\[ dQ=a_v\frac{n-k}{\,n-1\,}\,dT+bT\,dT; \]

whence

\[ \frac{dQ}{A\,dL} = \frac{n-k}{1-k} + \frac{b(n-1)}{a_v(1-k)}\,T = \alpha+ \frac{b(n-1)}{a_v(1-k)}\,T \qquad \ldots \qquad (46) \]

The dependence $Q=f(L)$ is already obtained in a complicated form. Since from (45), taking the initial temperature as $T_0$,

\[ T=T_0+\frac{n-1}{1-k}\,\frac{AL}{a_v}, \]

then

\[ dQ = A\frac{n-k}{1-k}\,dL + A\frac{b(n-1)}{a_v(1-k)}\,T_0\,dL + b\left[\frac{A(n-1)}{a_v(1-k)}\right]^2 L\,dL, \]

and

\[ Q=DL+EL^2,\qquad \ldots \qquad (47) \]

which gives a parabola in the coordinates $QL$.

Similarly, one may find the relation between $L$ and $\Delta U$, and also between $\Delta U$ and $Q$, the latter being still more complicated.

Let us now proceed to the consideration of the process in the coordinates $TS$. From equation (35a) it is clear that

\[ S=g\ln T+bT+\mathrm{const.}\qquad \ldots \qquad (47) \]

This equation gives logarithmic curves, which run quite differently depending on the sign of the quantity $g$. For $g<0$ these curves ...

intersect the abscissa axis at \(S=\infty\), and at \(T=\infty\) they touch the straight lines \(MT+\) const. A series of polytropes with exponent \(n=1.32\) (air) is given in Fig. 14.

According to what has already been said, at

\[ T_m=-\frac{g}{b} \]

we obtain the point dividing the curve into parts with heat supplied and heat removed. The tangent at this point is perpendicular to the \(S\)-axis. In the example shown in the drawing, air, when expanded to \(T_m=1107^\circ\), loses heat; with further expansion heat is supplied to it. What the position of the polytrope with the same exponent but with \(C_v=\mathrm{const.}\) would be relative to the axes \(T\) and \(S\) is shown by the dotted curve. Processes analogous to \(ED\), when \(T_{\max}-T_m=T_m-T_{\min}\), give, as was indicated above, \(Q=0\), i.e. the areas shaded at the extreme right and left curves are equal to one another. With \(C_v=\mathrm{const.}\), the process, for the same temperature difference, would be represented by the curve \(E_1D_1\). The area under the curve shows without further explanation how impermissible it is, in calculating heat, to use a constant heat capacity. At \(g>0\), i.e. at \(n>k\), the heat exchange is obtained in one direction, as is seen from equation (47), and also from Fig. 15, where a series of polytropes with \(n=1.5\) (air) is given.

Summing up the consideration of the polytrope \(pv^n=\mathrm{const.}\) with variable heat capacity, one may conclude that this process can be interpreted as simply as in the case \(C_v=\mathrm{const.}\) only within very narrow limits. In general, such an interpretation leads to the determination of incorrect values for the quantities defining the final state of the body, and especially for the quantity \(Q\), the heat which the body exchanges during the process with the surrounding medium. Even the direction of heat exchange cannot be inferred from the value of the exponent \(n\); it is necessary to know the temperature limits between which the process takes place. Comparison of individual processes solely on the basis of the value of \(n\) is likewise impossible without knowing the limiting temperatures; therefore a definite change in the exponent \(n\) (if the curve of change may be regarded as a polytrope with variable exponent) may signify either an increase of heat exchange, or its decrease, or a change of direction, depending on the temperatures at which this change takes place. Moreover, just as for \(C_v=\mathrm{const.}\), even at the same temperature, a uniform change of \(n\) corresponds to a non-uniform change in the quantity \(C\), which most closely characterizes heat exchange. [(see (17)].

On estimating the change of the exponent \(n\)

In individual studies of the indicator diagrams of internal-combustion engines, the lines of compression and expansion are often characterized as polytropes with variable exponents. Let us consider one example. Rather much space was devoted to them, for example, in one of his works by F. Münzinger¹. Taking into account the dependence of heat capacity on temperature, he nevertheless operates with the exponent \(n\), in the qualitative evaluation of phenomena, almost in the same way as would be necessary for \(C_v=\mathrm{const.}\) He believes, pointing out that the difference between \(n\) and \(k^1\) (\(k^1\) is calculated for \(C_v\ne\mathrm{const.}\)) makes it possible to estimate the degree of influence of cooling, afterburning, etc. The exponents of the compression and expansion lines are reduced to diagrams, two of which are included in Figs. 16 and 17. Having the charge weight and the \(pv\) diagram, it was possible, from the characteristic equation, to find the temperatures which, for some points, are marked on the diagrams. The temperature must be known in order to determine the adiabatic exponent \(k\) corresponding to the investigated instant. Its value is determined from the expression:

\[ k^1=1+\frac{1.985}{mc_v}, \]

where \(m\) is the molecular weight of the mixture and \(C_v\) is the instantaneous heat capacity. Since the composition of the gas is not specified in the article, but several values of \(k\) at different temperatures are available, we use these data to find the quantities \(A\) and \(B\) in the expression:

\[ k^{1}=1+\frac{1.985}{A+BT}. \]

For the case shown in Fig. 16 (underload \(p_m=2.48\) atm.), we find their mean values, which makes it possible to express the heat capacity as:

\[ \begin{aligned} mC_v &= 4.76 + 0.000852\,T,\\ mC_p &= 6.745 + 0.000852\,T. \end{aligned} \tag{48} \]

Hence, for expression (37) and the others,

\[ k=\frac{a_p}{a_v}=1.417. \]

Assuming that separate moments of expansion may be regarded as instantaneous polytropic processes with a constant exponent \(n\), in studying the nature of heat exchange one must first of all find the value \(C\) in the expression:

\[ dQ=C\,dT=(g+bT)\,dT. \]

This quantity, which may be called the apparent instantaneous heat capacity, will give an idea of the intensity of heat exchange (without reference to time). The quantity \(g\) is determined from expression (37), where \(k\) has the value 1.417, \(a_v=4.76\), and \(n\) is taken from Table I (from the curve). The found values of \(C\) are given in Table I and plotted along the piston stroke in Fig. 16, as also are \(n\) and \(k^{1}\). From the drawing it is clear that even an approximate judgment of changes in heat exchange from the difference \(n-k\) is difficult, and in the present example even impossible. In the first half of the piston stroke, the actual increase in the difference \(n-k\) coincides with the increase in heat losses outward, but later the slow increase in the difference \(n-k\) is accompanied by a decrease in heat losses, and not by an increase, as was true under the same external circumstances for the first half of the stroke. In Fig. 16 there is given an indicator diagram of the same engine at a mean indicated pressure

Table I.

\(t^\circ\)	\(n\)	\(g\)	\(bT\)	\(g+bT\)	\(n-k\)
200	1.482	0.642	0.403	1.045	0.095
300	1.477	0.598	0.488	1.086	0.093
400	1.476	0.590	0.573	1.163	0.075
500	1.482	0.165	0.659	0.824	0.0675
600	1.395	\(-0.265\)	0.745	0.480	0.0375
700	1.355	\(-0.831\)	0.829	\(-0.002\)	0
800	1.3	\(-1.856\)	0.923	\(-0.933\)	\(-0.05\)

\(P_m=5.875\) atm. Here the difference \(n-k\) slowly grows with increasing piston stroke. A calculation analogous to the preceding gives:

\[ \begin{aligned} m c_v &= 4.106 + 0.001888\,T,\\ m c_p &= 6.091 + 0.001888\,T,\qquad k=1.485. \end{aligned} \]

A rough estimate from the exponent \(n\), or from the difference \(n-k\), would give incorrect results. From Table II and from curve \(C\) in Fig. 16 it is seen that the exponent \(k\) and the difference \(n-k\) increase toward the end of expansion; heat losses outward decrease, and quite considerably. Thus from the two examples given it is clear how cautiously one must proceed in using the difference \(n-k\) to judge heat exchange. The conclusion from the given examples that the increase of \(n-k\) toward the end of expansion is a consequence of the influence of the increasing cooling surface would likewise be incorrect, since this conclusion implies an increase in losses, which in fact is not present. Further, even comparison of individual diagrams, and of individual points in them, by the difference \(n-k\) is made difficult not only by the fact that one is comparing

moments at different temperatures, but also because separate diagrams, taken at different loads, are obtained for different mixtures; therefore identical differences \(n-k\) at identical temperatures do not give identical \(C\)¹).

Table II.

\(t\)	\(n\)	\(g+bT\)
600	1.433	1.157
700	1.421	1.247
800	1.410	1.275
900	1.398	1.316
1000	1.380	1.270
1100	1.370	1.317
1200	1.364	1.115
1300	1.356	1.481

The errors of such an estimate increase still more when, in computing \(k\), the heat capacity is assumed constant. The matter is still further complicated when the quantity \(n\) is replaced by the magnitude of the angle \(\psi\), the angle of inclination of one of the straight lines passing through the origin that are needed in constructing the polytrope by Brauer’s method. This is the basis of Leinweber’s “diagram characteristic”²). Its essence may be seen from Fig. 18. The curve of change \(ABC\ldots\) is given in the coordinates \(pv\). Through the origin we draw a straight line inclined to the axis of abscissas at an angle \(\varphi\). By a vertical through the point \(A\) we intersect this inclined line, and through the point of intersection we draw a straight line inclined to the \(v\)-axis at \(45^\circ\), up to its meeting with this axis (as also in the construction according to Brauer). The vertical through the point of meeting intersects the curve of change at the point \(B\). Further, by the construction indicated in the drawing, we find the point \(b\), which, together with the origin, determines the position of the straight line inclined at an angle \(\psi\) to the axis of ordinates. This will be the second straight line needed in the construction of the polytrope, the angle \(\psi\) for which is found from the expression:

\[ 1+\operatorname{tg}\psi=(1+\operatorname{tg}\varphi)^n . \tag{49} \]

In our case, however, \(\varphi\) and \(\psi\) are known, and thus from this expression one can find the exponent \(n\), which will be the mean value for the segment \(AB\). Repeating similar constructions further, we obtain a series of points \(b, c, d\ldots\), determining the position of the characteristic. If one also plots the straight lines corresponding to the exponents \(n=1\) and \(n=k\), then the course of the characteristic between them will make it possible to estimate the direction of the change of \(n\).

The intersection of the characteristic with the straight line \(n=k\) will precisely determine the segment containing the point dividing the curve into parts with removal and addition of heat. The shortcomings of the characteristic (apart from drafting ones) consist in the fact that the relation between \(n\) and \(\psi\) (49) is not so simple that, from the change of \(\psi\) along the characteristic, one could infer how rapidly \(n\) changes; moreover, its construction and interpretation are simple only for \(C_v=\text{const.}\), the admission of which can give the greatest errors. In order to introduce a variable \(k\), it is necessary, by computation, to find it from (49) and to plot a curve instead of the straight line \(n=k\). This deprives the method of simplicity and, at the same time, still gives means for interpreting heat exchange that are less perfect than in the cases analyzed above (Figs. 16 and 17), since it includes in the consideration an extra link \(\psi\), whose direct relation to the quantities \(C\), or \(Q\), is too complicated.

On the meaning of the equation \(pv^n=\mathrm{const.}\)

Technical thermodynamics, while giving general laws and general rules for the investigation of thermal processes, also gives ready-made results of investigations for certain particular cases. Of the infinitely great number of processes, these are the simplest, but they are of great significance in the study of actual, real processes, since they make it possible to compare the latter with ideal ones, i.e., with those limiting ones to which it is desirable to bring real ones nearer, the course of which we can partly influence. Among such simple (basic)

¹) Taking this into account, one can obtain somewhat different results from Fig. 24 in Münzinger.

²) Leinweber, “Diagramm-Charakteristiken.” Z. d. V. d. I. 1913.

The processes \(p = \mathrm{const.}\), \(v = \mathrm{const.}\), \(T = \mathrm{const.}\), and \(dQ = 0\) belong to the processes. A formal study of them shows that they are special cases of a general process, whose law of progression, at constant heat capacity, can be represented as \(pv^n = \mathrm{const.}\)

In a real setting, the process that is fundamental in idea is invariably accompanied by a number of secondary irreversible processes, owing to which the process taken as a whole is also irreversible. For an infinitely small change in a gaseous medium this is expressed as follows:

\[ dQ' = dW + dU + Ap\,dv, \]

where \(dW\) will represent the differential of the energy converted into heat by friction, or lost as a result of radiation, thermal conduction, transformed into the energy of vortices, etc. In practice, in our instruments it is often necessary to obtain such processes in which no work is performed and all the communicated heat passes into the thermal energy of the working body. In that case we obtain the process \(v = \mathrm{const.}\), for which

\[ dQ' = dW + dU. \]

The difference from the ideal case consists only in the fact that it is irreversible. But formally it will be similar to the ideal one if we write

\[ dQ' - dW = dQ = dU \]

and consider it only in one direction. The same relations between the ideal and the real will hold in the case \(p = \mathrm{const.}\), when

\[ dQ' - dW = dQ = dI, \]

where \(I\) is the symbol of the so-called heat content of the working body. The meaning of studying the ideal processes \(p = \mathrm{const.}\) and \(v = \mathrm{const.}\) is that all real ones have the same form of study.

Of the enormous number of cases of distribution of the heat supplied during a process, the case is undoubtedly of interest in which all heat is converted into work, i.e. when (gases)

\[ dQ = Ap\,dv, \]

which gives \(T = \mathrm{const.}\),

and the case when work is performed by the body without heat being supplied to it from outside, i.e. when \(dQ = 0\). These processes, first, make it possible to obtain the most perfect heat-saving ideal cycle and, second, are of very great importance as the best desirable models for separate parts of real cycles. How heat is distributed in an actual real process is a question that is both difficult and unresolved for our heat engines. One may imagine that the study of all circumstances of the work of bodies can give an exact picture of the distribution of \(Q\) among \(U\), \(AL\), and \(W\) for each moment in the process. Clarification of such a picture would make it possible to establish the limiting, best process for an individual given case, which may not coincide with what is commonly ideal in thermodynamics, since it also accounts for the influence of real conditions. The laws of distribution of \(Q\), however, are so complex and as yet so unknown that it is impossible to make use of them. A way out is provided by simplified assumptions. Thus the assumption that heat at every moment passes, in one definite constant part, into work gives, for \(C_v = \mathrm{const.}\), the simple law of progression of the process \(pv^n = \mathrm{const.}\); this assumption, if applied to the process in an engine, stands up to criticism so well that the widespread use of the equation \(pv^n = \mathrm{const.}\), and the efforts to replace the true curve by an approximate one according to the same equation, must evidently be explained by it. First of all, as has already been mentioned, this equation is common in form with the equations for the simplest fundamental processes. Then, in fact, with small temperature changes, the curves of true processes can often be replaced, with a close approximation, by the curves-vesm

$p v^n = \mathrm{const}.$ The equation $p v^n = \mathrm{const}.$ is comparatively simple and, as a result, convenient for use in calculations. Finally, from the exponent $n$, when $C_v = \mathrm{const}.$, one can draw certain conclusions about heat exchange and about the rate of pressure drop. Owing to all this, the tendency has become widespread to characterize individual lines of the real process in an engine by a polytrope with some average value, for each type or set of conditions, of its exponent. If such a value of $n$ has been established more or less reliably, then it makes it possible, approximately, in preliminary computations, to depict the process also for the machine being designed. In this lies almost the entire practical significance of the polytrope. It does not have the significance possessed by the theoretical processes $p = \mathrm{const}.$, $v = \mathrm{const}.$, $T = \mathrm{const}.$, and $dQ = 0$. This curve is undoubtedly useful so long as the processes being studied are those in which its use introduces simplifications at the cost of small inaccuracies (for example, in air engines and compressors); but of course, if it has not been proved that the thermal processes occurring in the machine must give the law of their course in the form $p v^n = \mathrm{const}.$, the use of this curve should be regarded only as an auxiliary device whose purpose is to alleviate somewhat the external difficulties of the calculations. That the inner essence of a process formerly regarded as polytropic

\[ \left(\frac{\Delta U}{L} = \mathrm{const.}\right) \]

is of no importance is evident from the fact that, when there is an obvious need to take account of the variability of heat capacity, equation (25) is not used, while the former $p v^n = \mathrm{const}.$ is retained, which must already be considered a relic of the past, a relic for which, in some cases, there is as yet nothing reliable to substitute. In investigations of real thermal processes, even the equation $p v^n = \mathrm{const}.$ is already losing its significance. The methods employed in study may not require the determination of polytropic exponents for individual segments or points of the curve of change of state.¹ Most of them have as an auxiliary tool the entropy diagram (also the heat diagram, and also Professor Berstol’s energy diagram); these methods are graphical. The work of investigation is placed on still firmer ground when the complex thermal phenomena in the cylinder of a machine are separated and studied individually, resting on solid physical foundations.² Such study makes it possible to accumulate experimental material of a more general character than that which is available in the form of a known mean exponent for the line of change in a machine of a strictly definite type, and, most importantly, one that more deeply elucidates the inner essence of the process itself.

A. V. Krasnikov, “Heat Transfer in a Polytropic Process”

Fig. 1.
Visible labels: \(p\), \(V\), \(A\), \(B\), \(T_1\), \(T_2\), \(T_0\), \(V_1\), \(V_2\), \(V_0\), \(A_1\), \(A_2\), \(B\), \(n\), \(V_1\), \(V_2\).

Fig. 2.
Visible labels: \(p\), \(V\), \(A\), \(B\), \(C\), \(S\), \(T_1\), \(T_2\), \(T_{a1}\), \(T_{a2}\), \(q\), \(q_1\), \(q_2\), \(n=1\), \(n=0\), \(n=\infty\), \(C_v\), \(C_p\), \(\Delta Q\), \(\Delta Q_2=[T_2-T_0]C_v\).

Fig. 3.
Visible labels: \(n\), \(0\), \(K\), \(C_v\), \(C_p\), \(\dfrac{\varphi Q}{dT}\), \(V\).

Fig. 4.
Visible labels: \(T_1\), \(T_2\), \(A\), \(B\), \(t_0\), \(M\), \(M_{\mathrm{pol}}\).

Fig. 5.
Visible labels: \(p\), \(V\), \(A\), \(B\), \(C\), \(C'\), \(n\), \(T\), \(dT\), \(dV\), \(A\delta q,V\), \(p\), \(n=0\), \(n=1\).

Fig. 6.
Visible labels: \(Q\), \(A\), \(B\), \(C\), \(O\), \(V\), \(Q=c(T)\) [[unclear: formula partly obscured]].

Fig. 7.
Visible labels: \(p\), \(V\), \(A\), \(B\), \(C\), \(A\delta q,V\), \(n=1\).

Fig. 8.
Visible labels: \(p\), \(V\), \(A\), \(B\), \(A\delta q,V\), \(n=\infty\), \(n=1\), \(C\).

![Diagram with thermodynamic curves and axes labeled \(V\), \(S\), \(T\), \(P\); curves and points labeled \(A\), \(B\), \(C\), \(D\), \(I\), \(II\), \(III\), \(T=C\), \(\delta T\), \(R\), and other symbolic annotations.]

Fig. 6.

![Diagram with thermodynamic curves and axes labeled \(V\), \(S\), \(T\), \(P\); curves and points labeled \(A\), \(B\), \(I\), \(II\), \(III\), \(T=C\), \(\delta T\), \(R\), and other symbolic annotations.]

Fig. 8.

![Diagram with thermodynamic curves and axes labeled \(T\), \(Q\); points and segments labeled \(E\), \(W_1\), \(W_2\), \(\delta T\), \(T_1\), \(T_2\), \(T_3\), and other symbolic annotations.]

Fig. 9.

![Diagram with thermodynamic curves and axes labeled \(T\), \(Q\); curves and points labeled \(K\), \(T_m\), \(T_1\), \(T_2\), \(T_3\), \(T=1\), \(r=1\), \(1255^\circ\), \(0\), and other symbolic annotations.]

Fig. 10.

U. N. Kaverin, “Heat Transfer in a Polytropic Process”

Fig. 11.

Visible labels in the figure: \(T\), \(U\), \(n=1\), \(n=1.13\), \(n=1.35\), \(n=1.45\), \(n=1.4\), \(n=1.72\), \(k=1.41\), \(c_v=\infty\), \(c_v=0\), \(c_v=-c_0\), \(c_v\to-\infty\).

Fig. 12.

Visible labels in the figure: \(Q\), \(T\), \(T_1\), \(p\), \(v\), \(a\), \(b\), \(c\), \(d\), \(e\), \(T_2\), \(b'\), \(c'\), \(d'\), \(e'\), \(c_v=0\), \(c_v<0\).

Fig. 13.

Visible labels in the figure: \(Q\), \(T\), \(T_1\), \(T_2\), \(a\), \(b\), \(c\), \(d\), \(e\), \(b'\), \(c'\), \(d'\), \(e'\), \(c_v=\infty\), \(c_v<0\), \(p\), \(v\).

Fig. 14.

Visible labels in the figure: \(T\), \(U\), \(S\), \(2000^\circ\), \(T_k=\dfrac{a}{b}\), \(a\), \(b\), \(c\), \(d\), \(D\), \(C\), \(B\), \(A\), \(Q\), \(n=1.28\).

Fig. 15.

Visible labels in the figure: \(T\), \(2000^\circ\), \(bT\), \(A\), \(B\), \(C\), \(D\), \(Q\), \(S\).

Fig. 16.

Visible labels in the figure: \(T_{\max}\), \(Q\), \(0.2\), \(40\), \(60\), \(800^\circ\), \(900^\circ\), \(1000^\circ\), \(1100^\circ\), \(1200^\circ\), \(1300^\circ\), \(C\), \(n=1.4\).

Fig. 17.

Visible labels in the figure: \(S\), \(Q\), \(U\), \(q\).

Fig. 18.

Visible labels in the figure: \(Q\), \(20\), \(40\), \(60\), \(80\), \(700^\circ\), \(600^\circ\), \(500^\circ\), \(400^\circ\), \(300^\circ\), \(200^\circ\), \(C\), \(bT\), \(1.2\), \(1.3\), \(1.4\), \(1.5\).

1. Weisshaar, “Untersuchungen über den Verlauf der Verbrennung im Dieselmotor,” Z. d. V. d. I., 1916. Neumann, “Untersuchungen an der Dieselmaschine,” Forschungsarbeiten, Heft 245. Wilkins, “Trials on a Diesel Engine,” The Journal of the Inst. of Mech. Eng., 1916, and others. ↩↩↩

2. Nusselt, “Der Wärmeübergang in der Verbrennungskraftmaschinen,” Z. d. V. d. I., 1914 and 1923. Eichelberg, Forschungsarbeit, Heft 263. ↩