On the Construction of a Universal Thermal Diagram of an Ideal Gas

G. I. Fuks

I

A new type of universal thermal diagram for an ideal gas is proposed, the construction of which is based on the relations:

\[ \left(\frac{\partial i}{\partial T}\right)_p=c_p, \tag{1} \]

\[ \left(\frac{\partial s}{\partial T}\right)_p=\frac{c_p}{T}, \tag{2} \]

\[ \left(\frac{\partial s}{\partial T}\right)_v=\frac{c_v}{T}. \tag{3} \]

The last two relations may be rewritten as follows:

\[ \left(\frac{\partial s}{\partial \ln T}\right)_p=c_p, \tag{4} \]

\[ \left(\frac{\partial s}{\partial \ln T}\right)_v=c_v. \tag{5} \]

From (1), (4), taking into account the geometrical meaning of the derivatives, it follows that, with a suitable choice of scales, the curves

\[ i=f_1(T) \]

and

\[ s=f_2(\ln T), \]

and the tangents to them at a given temperature will be parallel. On the diagram one may plot a scale for \(C_p\). Then the direction of the tangent will be found if the point on the scale line is connected with the origin of coordinates.

II

On the diagram, temperature is laid off along the ordinate axis on the left; along the abscissa axis: at the top—entropy \(\frac{\text{e}}{\text{n.m}^3}\), at the bottom—enthalpy \(i\ \frac{\text{kcal}}{\text{n.m}^3}\). On the ordinate axis on the right are laid off the natural logarithms of the temperatures¹. In addition, a scale for \(C_p\) is plotted.

For reading off the change in internal energy and for plotting the lines

\[ s_v=f_3(\ln T) \]

an inclined straight line in the left-hand part of the diagram is used. Indeed, as is known, for ideal gases

\[ i-u=ART, \tag{6} \]

which for \(\mathrm{nm}^3\) \(\left(R=37.85\,\dfrac{\mathrm{kg\,m}}{\mathrm{nm}^3\,\mathrm{deg}}\right)\)

gives

\[ i-u=0.0886T. \]

For entropy at \(p=\mathrm{const}\) and \(v=\mathrm{const}\), respectively, we have:

\[ ds_p=c_p\frac{dT}{T}, \]

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

whence

\[ d(s_p-s_v)=(c_p-c_v)\frac{dT}{T}=AR\frac{dT}{T}. \tag{8} \]

Integrating, we obtain for \(\mathrm{nm}^3\)

\[ s_p-s_v=0.0886\ln T, \tag{9} \]

which is a straight line in the coordinates \(\ln T\)—\(s\), coinciding, with the adopted scale ratio, with the straight line \(i-u\) in the coordinates \(i\)—\(T\).

For transition to different pressures and volumes, a logarithmic scale is plotted for \(\dfrac{p_2}{p_1}\left(\dfrac{v_2}{v_1}\right)\). The segments of this scale give, as usual, the magnitude of the shift when passing to other \(p\) and \(v\) (shift to the right for \(p_2<p_1\) and \(v_2>v_1\)).

For convenience of calculations, the diagram includes a table of enthalpies of ideal gases¹. From enthalpy it is easy to pass to the heat capacity \(c_p\). Usually, however, this is not necessary, since it is simpler to use the enthalpy values directly.

With constant heat capacities \(c_p\) and \(c_v\),

\[ i=f_1(T) \]

and

\[ s_p=f_2(\ln T) \]

are represented on the proposed diagram by parallel straight lines; in other words, by one and the same straight line, along which the reading \(\Delta i\) is taken, within the temperature limits, on the left-hand scale, and the reading \(\Delta s_p\) on the right-hand scale. The line

\[ s_v=f_3(\ln T) \]

will also be a straight line. It is easy to construct it, since the straight line in the left-hand part of the diagram gives the difference \(s_p-s_v\) for any temperature interval. Thus, with constant heat capacity, the thermal diagram will consist only of straight lines, i.e., it has the simplest form.

In most cases, quite sufficient accuracy is obtained by calculating with the mean heat capacity \(c_p\) for the given temperature interval; in other words, by regarding the change of \(i\) over the calculation temperature interval as linear. This same straight line gives the change of \(s_p\), and consequently the diagram will likewise consist of straight lines.

In more accurate calculations, one may plot, using an enthalpy table, several intermediate values of \(i\). By connecting them with straight lines and drawing lines parallel to them in the corresponding intervals of the logarithmic temperature scale, we find the corresponding curve

\[ s_p=f_3(\ln T). \]

Next, using the straight line in the left-hand part of the diagram, one plots

\[ s_v=f_3(\ln T). \]

The curves for air (\(i\) and \(s_{pr}\)) are plotted on the diagram.

Comparison of the proposed diagram with known universal diagrams for ideal gases leads to the following conclusions:

1. The Stodola–Zeuner diagram is less accurate, since it assumes an artificial dependence of the heat capacity \(C_v\) on temperature in the form of a linear function with the same constant term for all gases.

2. The Schüle diagram requires a considerable amount of preliminary computation in order to obtain the lines \(s_p\) and \(s_v\) for a specified composition of a gas mixture. As heat capacities are refined, the principal lines of the Schüle diagram must be reconstructed.

In refined calculations, taking account of variable heat capacity, it is likewise necessary in the proposed diagram to construct auxiliary curves for \(i\) and \(s_p\). In this case, however, it turns out that one can make do with a small number of intermediate points. The results of calculating the errors in the value \(\Delta s_p\) for various gases with different numbers of intermediate points are presented in the following table.

Table 1

The data in the table show that 2–3 intermediate points ensure an accuracy of up to 1% in any temperature interval of the diagram. The intermediate points are best chosen by dividing the temperature interval, in logarithmic coordinates, into equal parts.

The disadvantages of the diagram under consideration should be taken to be:

1. The nonuniform (logarithmic) temperature scale.

If a sufficient number of intermediate divisions for temperatures is plotted, this cannot be of any special importance.

1. The impossibility of measuring heat by areas, as in the natural \(T\)—\(s\) diagram.

In practice, as is known, this method is usually not used. For a polytropic process, for example, it is usually recommended to calculate the heat from the relation

\[ Q=T_c\Delta s, \tag{10} \]

where \(T_c\) is the mean temperature, usually the arithmetic mean. An analogous calculation with greater accuracy can be carried out using the proposed diagram—

As is known, for any process with constant heat capacity the mean thermodynamic temperature is equal to the logarithmic mean

\[ T_{cm}=\frac{T_2-T_1}{\ln \frac{T_2}{T_1}} . \tag{11} \]

The value of the denominator is easily read off on the left ordinate axis of the diagram. On the other hand, every process with constant heat capacity in the coordinates \(\lg T—S\) is represented by a straight line. Replacing an arbitrary process by a broken line, one can determine the mean temperatures and the heat for individual sections.

Fig. 2

If, in order to estimate the mean temperature, it is read off on the right ordinate axis at the middle between the initial and final temperatures of the process, then we find the geometric-mean temperature

\[ T=\sqrt{T_1T_2}, \tag{12} \]

which differs less from the mean thermodynamic temperature than the arithmetic mean.

Fig. 3

Figures 2 and 3 show the execution of a numerical example of a polytropic expansion process with exponent \(n=1.32\), from \(p_1=60\) ata, \(t_1=2000^\circ\mathrm{C}\) to \(p_2=3\) ata, of a gas having the following composition by volume:

\[ 80\%\,\mathrm{N_2},\ 15\%\,\mathrm{CO_2}\ \text{and}\ 5\%\,\mathrm{H_2O}. \]

Figure 2 shows the determination of the final temperature. Since

\[ n=\frac{\overline{cb}}{\overline{ca}} =\frac{\overline{cb}}{\overline{cb}-\overline{ab}}, \quad \text{then} \quad \overline{ab}=\frac{n-1}{n}\,\overline{cb}. \]

On the pressure scale, for \(\dfrac{p_2}{p_1}=\dfrac{60}{3}=20\), we have \(\overline{cb}=123.7\) mm.

Therefore,

\[ \overline{ab}=\frac{1.32-1}{1.32}\,132.7=32.2\ \text{mm}. \]

We lay off from the straight line \((s_p-s_v)\) to the left

\[ b_1a_1=\overline{ba} \]

and draw \(a_1d_1\) vertically. This gives \(t_2=830^\circ\text{C}\). Taking into account the composition of the gas, from the enthalpy table we have

\[ \text{At } t_1=2000^\circ\text{C}, \qquad i_1=789.8\ \frac{\text{kcal}}{\text{nm}^3}, \]

\[ t_2=830^\circ\text{C}, \qquad i_2=298.5\ \frac{\text{kcal}}{\text{nm}^3}. \]

In the coordinates \(i-t\), the corresponding points are marked and the straight line (Fig. 3) \(\overline{ab}\) is drawn.

The straight line \(AB\) in the coordinates \(\lg T-s\) gives the value \(\Delta s_p\). The change in internal energy

\[ \Delta u=-386.5\ \frac{\text{kcal}}{\text{nm}^3} \]

is counted off by drawing \(\overline{ac}\) parallel to the straight line \((i-u)\) of the diagram. The polytrope in thermal coordinates is represented by the straight line \(AC\), which gives

\[ \Delta s=-38\cdot 10^{-3}\ \frac{\text{kcal}}{\text{nm}^3}. \]

The mean temperature according to (11) is

\[ T_{\text{m}}=\frac{1170\cdot 200}{144.5}=1619^\circ\text{K}, \]

where \(144.5\) mm is the vertical distance between \(A\) and \(C\), and \(200\) is the value of the logarithmic scale.

The heat of the process is

\[ q=T\Delta s=-1619\cdot 38\cdot 10^{-3}=-61.6\ \frac{\text{kcal}}{\text{nm}^3}. \]

If the work is calculated from the relation

\[ Aw=\frac{AR}{n-1}(t_1-t_2), \]

then one obtains

\[ Aw_n=\frac{0.0886}{1.32-1}\,1170=323.9\ \frac{\text{kcal}}{\text{nm}^3}. \]

Then

\[ q=\Delta u+Aw \]

will be

\[ q=-386.5+323.9=-62.6\ \frac{\text{kcal}}{\text{nm}^3}. \]

\(i\)-\(T\)-\(S\) Diagram for Ideal Gases

Top scale: \(S\), kcal/\(\mathrm{Nm}^3\cdot{}^\circ\mathrm{C}\)
Left and right vertical scales: temperature, \({}^\circ\mathrm{C}\)
Bottom scale: \(i\), kcal/\(\mathrm{Nm}^3\)

Scale annotations:

• Enthalpy: \(1\ \dfrac{\mathrm{kcal}}{\mathrm{Nm}^3} \sim 0{,}5\ \mathrm{mm}\)
• Entropy: \(1\ \dfrac{\mathrm{kcal}}{{}^\circ\mathrm{C}\,\mathrm{Nm}^3} \sim 500\ \mathrm{mm}\)
• Temperature: \(1^\circ\mathrm{C} \sim 0{,}2\ \mathrm{mm}\)
• Direction of the temperature logarithms: unit \(\sim 200\ \mathrm{mm}\)

Curve labels and scale labels visible in the diagram:

• \(i-u\)
• \(S_p-S_v\)
• \(S_p\)
• \(i_2\)
• \(C_p\), kcal/\(\mathrm{Nm}^3\cdot{}^\circ\mathrm{C}\)
• \(\dfrac{p_2}{p_1}\left(\dfrac{v_2}{v_1}\right)^k\)

Enthalpy of Gases in kcal/\(\mathrm{Nm}^3\)

1. According to the data of Chernobaev and Zhivotovskii. ↩↩