DEGREE OF PERFECTION OF A THERMAL POWER PLANT

G. I. Fuks

1

The generally accepted method of assessing the degree of perfection of a thermal power plant by its efficiency according to the relation

\[ \eta=\frac{860N_{\mathrm{e}}+\sum Q_{\mathrm{u}}}{BQ_{p}} \tag{1} \]

is not sufficiently justified. If we have two plants that consume the same quantity of the same fuel and deliver the same quantity of electric power and heat, then they have identical efficiencies, even if the parameters of the heat delivered by each plant are different. In a back-pressure plant with complete utilization of the exhaust steam, a deterioration of the internal relative efficiency of the turbines may produce an increase in the efficiency of the plant; and if the turbine is replaced by a throttling valve, the plant may attain the maximum possible value of the efficiency, and so forth. Since these and similar conclusions contradict common sense, several ways of circumventing them have been proposed.

In the work of Professor I. N. Butakov [1], the value of the efficiency of a thermal power plant is made dependent on the value of the efficiency of a purely condensing plant, \(\eta_k\), and of a purely back-pressure plant, \(\eta_t\). It turns out that, with an increase in the generation of electric power for heat consumption, the efficiency of the plant increases from \(\eta_k\) to \(\eta_t\), which is taken by the author as proof of the fundamental correctness of [1].

But in order to obtain the corresponding result, the author had to introduce the assumption that the various plant variants are compared with one another at the same generation of electric power, the deficiency of the latter being covered by an additional condensing plant with efficiency \(\eta_k\). The very values \(\eta_k\) and \(\eta_t\) for a given thermal power plant with efficiency \(\eta\) are conditional quantities. Outside these or analogous arbitrary conditions, in essence, it is impossible to obtain a substantiated proof of the correctness of relation (1) or to remove the contradictions indicated above.

The method of characterizing the degree of perfection of a thermal power plant by the magnitude of the specific generation of electric power for heat consumption has come into wide use [2], [3]. Indeed, the specific generation of electric power for heat consumption depends on the quantities by which the degree of perfection of the plant is determined. For a back-pressure plant, for example, on the basis of this indicator we require the maximum increase in the relative efficiencies of the turbine, which agrees with the requirement of economy, and so on. But this indicator by itself, as a number, is of course insufficient for judging the degree of perfection of the plant. Thus, in the limiting cases of purely condensing

and for purely heating installations it assumes the limiting values \(\infty\) and 0, which is not connected with their degree of perfection. The use, in assessing the degree of perfection of the installations being compared, of two indicators together—namely, efficiency and the specific generation of electric power per heat consumption—is inconvenient because these quantities may vary in opposite directions. If, for example, two installations being compared consume the same amount of the same fuel and deliver the same amount of electric power and heat, but of different potentials in the latter case, then these installations will have the same efficiencies and the same values of specific generation of electric power per heat consumption. But it is clear that the installation delivering heat of a higher potential will, under the given conditions, be more perfect. Other examples may also be cited in which both of the indicated indicators taken together do not solve the problem posed.

Professor M. M. Khazen [4], [5] proposes comparing the operation of a thermal power installation with that of a certain reference installation, for which the degree of perfection is conventionally taken as 100%. This proposal met with a number of objections [6], [1], in which the arbitrariness of the reference case and of the method of comparison with it was rightly pointed out. Nevertheless, it must be acknowledged that M. M. Khazen made the first attempt to create such a method for evaluating the degree of perfection, one that goes beyond the customary range of notions concerning the identity of this quantity with efficiency.

Sometimes one encounters the assertion that it makes no sense to compare installations delivering heat of different potentials, since they operate under incomparable conditions. Such a refusal to address the posed problem cannot be regarded as its solution.

If in a power system we have stations that deliver heat of different potentials, then, without being able to compare them with one another, it is impossible to distribute the load correctly among them. In principle, all forms of energy represent different forms of motion, whose transformation is subject to definite laws. Having “expended” energy in one form and “received” it in another, we are always entitled to pose the question of how perfectly this transformation has taken place. It is only necessary to be able to find the proper method of comparison.

The relation for efficiency (1), in general form, may be expressed symbolically as follows:

\[ \eta=\frac{E_{\text{used}}}{E_{\text{exp}}}, \tag{2} \]

where \(E_{\text{used}}\) and \(E_{\text{exp}}\) are the energy used and expended, respectively. Efficiency according to (2) is therefore considered a measure of the degree of perfection because it is assumed that

\[ E_{\text{exp}}-E_{\text{used}} \]

gives the amount of energy that was “lost” in its transformations. But in reality the latter condition is not always satisfied: thus, the efficiency of a heat engine delivering work \(AL\) kcal with an expenditure of heat \(Q_1\) will be

\[ \eta_i=\frac{AL}{Q_1}. \tag{3} \]

This assumes that the lost work is

\[ Q_2=Q_1-AL \text{ kcal}. \]

But in reality this is not so. If we set aside the hypothetical cases of supplying heat to the working body of a machine at an infinitely high temperature and removing heat at absolute zero, then in a heat engine, in the absence of any losses, it is impossible to obtain \(Q_1\) kcal of work. This means that, in this case, the efficiency according to (3) gives the fraction of heat converted into work. The degree of perfection of this installation, however, will obviously be expressed by a number greater than the value of its efficiency.

Let, in a heat exchanger of the usual type, after expending heat \(Q_1\) kcal, we transfer to the body being heated \(Q_2\) kcal. The efficiency of the heat exchanger is

\[ \eta = \frac{Q_2}{Q_1}. \tag{4} \]

This is based on the fact that the loss is taken to be

\[ Q_1 - Q_2 \ \text{kcal}. \]

In a heat exchanger of the usual design this loss is in reality greater, since heat is expended at a higher temperature than it is received. If the transfer of heat from a body with temperature \(T_1\) to a body with temperature \(T_2\) \((T_1 > T_2)\) is carried out without losses, i.e. by quasistatic processes (reversibly), for example by means of a heat pump, then more heat can be obtained than was expended, using the practically unlimited reserve of thermal motion of the surrounding medium. In this case as well, the efficiency according to (4) will give the fraction of heat transferred from the first body of the heat exchanger to the second, but the degree of perfection of the heat exchanger will be expressed by a number smaller than its efficiency.

Thus we arrive at the conclusion that expressions for the efficiency of an installation, composed on the basis of the fundamental relation (2), give the numerical value of the fraction of energy conversion, but do not always give an assessment of the degree of its perfection. The degree of perfection of an installation may be greater or smaller than its efficiency, depending on the quality of the energy expended and used.

To assess the degree of perfection in all energy conversions, one should use the fundamental relation

\[ \mu = \frac{E_{\text{used}}}{E_{\text{max}}}, \tag{5} \]

where \(E_{\text{max}}\) is the maximum amount of energy in the form used that can be obtained at the expense of \(E_{\text{exp}}\) by the most advantageous path of energy conversion. Indeed, the difference

\[ E_{\text{max}} - E_{\text{used}} \]

always gives the “loss” of the actual process of energy conversion, which justifies (5).

It should be emphasized that the magnitude of the degree of perfection\(^1\) \(\mu\), calculated according to (5), is fundamentally different from the efficiency \(\eta\), calculated according to (2). It is possible that

\[ \mu \begin{gathered} <\\ > \end{gathered} \eta . \]

The possibility of their equality is also not excluded. Thus, it is not difficult to show that in purely mechanical energy conversions, in the conversion of electrical energy into mechanical energy, etc.,

\[ E_{\text{max}} = E_{\text{exp}}, \]

and consequently the values of \(C\) and of the efficiency coincide; but a mechanical generalization of this proposition to all cases is incorrect.

\(^1\) Hereafter denoted everywhere, in abbreviated form, by \(C\).

It is advisable to begin the compilation of expressions for a C plant with the simplest cases. Let, in a heat engine, heat \(Q_1\), kcal, be supplied to the working body in process \(AB\) (Fig. 1). This heat is measured by the area \(AB21\). If, in this process, the engine delivers work \(AL\), kcal, then

Fig. 1

\[ E_{\mathrm{use}} = AL, \tag{a} \]

\[ E_{\mathrm{exp}} = Q_1, \tag{b} \]

\[ \eta = \frac{AL}{Q_1}. \tag{3} \]

To calculate the maximum amount of work that can be obtained in this engine, one must take into account the conditions of its operation. If the temperature of the surroundings, into which heat can be discharged without limit, is \(T_0\), then the only possible most advantageous cycle of the plant will be \(ABbaA\) (see Fig. 1), the area of which measures \(E_{\max}\) in the present case. But

\[ \text{area } ABbaA = \text{area } AB21A - \text{area } ab21a, \]

or

\[ E_{\max} = Q_1 - T_0 \Delta S, \tag{6} \]

where \(\Delta S\) is the change in entropy of the working body in the process of supplying heat to it. On the basis of (5), the C of the plant will be

\[ \mu = \frac{AL}{Q_1 - T_0 \Delta S}. \tag{c} \]

A simple transformation gives

\[ \mu = \frac{AL}{Q_1\left(1 - \dfrac{T_0 \Delta S}{Q_1}\right)}. \tag{d} \]

The quantity \(\dfrac{Q_1}{\Delta S}\) has the geometrical meaning (Fig. 1) of the mean height of a rectangle equal in area to \(AB21A\) and with base \(\Delta S\). Physically, this quantity may be regarded as the mean thermodynamic temperature\(^1\) [7] of the supply of heat to the working body in process \(AB\):

\[ T_{\mathrm{mt}} = \frac{Q_1}{\Delta S}. \tag{7} \]

Consequently,

\[ \mu = \frac{AL}{Q_1\left(1 - \dfrac{T_0}{T_{\mathrm{mt}}}\right)}; \tag{8} \]

and, taking (3) into account,

\[ \mu = \frac{\eta}{1 - \dfrac{T_0}{T_{\mathrm{mt}}}}. \tag{9} \]

\(^1\) Hereafter abbreviated everywhere as C--TT.

If the power of the plant is \(N\) kW, and the hourly heat expenditure is \(Q_u\) kcal/h, then

\[ \mu=\frac{860N}{Q_u\left(1-\frac{T_0}{T_{cm}}\right)} \tag{10} \]

Expressions (8), (9), and (10) are solutions of the problem posed. For a machine operating on the Carnot cycle,

\[ T_1=T_{cm}, \tag{e} \]

since the \(C\)--\(TT\) of the isothermal process coincides with the actual temperature. Therefore

\[ \mu_k=100\%. \]

Under all conditions, \(C\) of a heat engine will be greater than its efficiency.

4

If in a heat exchanger, with a heat expenditure \(Q_1\) kcal, the heated body has received heat \(Q_2\) kcal, then

\[ E_{\mathrm{use}}=Q_2, \tag{a} \]

\[ E_{\mathrm{exp}}=Q_1, \tag{b} \]

\[ \eta=\frac{Q_2}{Q_1}, \tag{3} \]

To calculate the maximum quantity of heat that can be transferred to the heated body with a heat expenditure \(Q_1\) kcal, the real conditions of heat exchange must be taken into account. Let the hot body of the heat exchanger, in the process of heat exchange, change its state along \(1\)--\(2\) (Fig. 2), giving off

Fig. 2

heat \(q_1 \dfrac{\mathrm{kcal}}{\mathrm{kg}}\) (areas 1234), and the cold one along 5—6, receiving heat \(q_2 \dfrac{\mathrm{kcal}}{\mathrm{kg}}\) (areas 5678). Then

\[ Q_1 = u_1 q_1, \tag{c} \]

\[ Q_2 = u_2 q_2, \tag{d} \]

where \(u_1\) and \(u_2\) are the quantities of the heat-exchanging fluids. The maximum quantity of heat in the heated fluid can be obtained if all processes in the heat exchanger proceed quasistatically (reversibly). Under the same conditions of heat exchange it will then be possible to heat \(u_m > u_2\ \mathrm{kg}\), with the corresponding transfer of heat

\[ Q_m = u_m q_2. \tag{e} \]

The degree of perfection of the heat exchanger according to (5) is

\[ \mu = \frac{Q_2}{Q_m} = \frac{u_2}{u_m}. \tag{11} \]

The quasistatic processes in the heat exchanger may be represented as follows. The hot body operates in a heat engine according to the cycle \(21ab\), delivering the work

\[ Al_1 = q_1 - T_0 \Delta S_1\ \frac{\mathrm{kcal}}{\mathrm{kg}}, \tag{f} \]

and in all

\[ AL_1 = u_1 Al_1 = u_1 q_1 - u_1 T_0 \Delta S_1 = Q_1 - u_1 T_0 \Delta S_1. \tag{g} \]

Work is expended to set in operation the heat pump according to the cycle \(65cd\), with an expenditure of work for each \(\mathrm{kg}\) of working fluid

\[ Al_2 = q_2 - T_0 \Delta S_2\ \frac{\mathrm{kcal}}{\mathrm{kg}}. \tag{h} \]

Utilizing all the work of the engine \(AL_1\), one can heat \(u_m\ \mathrm{kg}\) with the transfer to it of heat \(Q_m\), i.e.

\[ AL_1 = u_m Al_2. \tag{i} \]

If \(u_2\ \mathrm{kg}\) operates in the heat pump, then the work expenditure will be

\[ u_2 Al_2 = u_2 q_2 - u_2 T_0 \Delta S_2 = Q_2 - u_2 T_0 \Delta S_2. \tag{k} \]

Taking these relations into account, we have from (11)

\[ \mu = \frac{u_2 Al_2}{u_m Al_2} = \frac{Q_2 - u_2 T_0 \Delta S_2}{Q_1 - u_1 T_0 \Delta S_1} = \frac{Q_2}{Q_1} \frac{ 1 - \dfrac{T_0}{\dfrac{Q_2}{u_2}\dfrac{1}{\Delta S_2}} }{ 1 - \dfrac{T_0}{\dfrac{Q_1}{u_1}\dfrac{1}{\Delta S_1}} }. \tag{l} \]

Use of (c), (d), (7) and (3) gives:

\[ \mu = \frac{Q_2}{Q_1} \frac{ 1 - \dfrac{T_0}{T''_{cm}} }{ 1 - \dfrac{T_0}{T'_{cm}} }, \tag{12} \]

\[ \mu = \eta \frac{1 - \dfrac{T_0}{T''_{cm}}}{1 - \dfrac{T_0}{T'_{cm}}}, \tag{13} \]

where \(T'_{cm}\) and \(T''_{cm}\) are the mean thermodynamic temperatures of heat removal from the hot body and heat transfer to the cold body, respectively.

In ordinary heat exchangers

\[ T'_{cm} > T''_{cm}, \]

therefore, \(\mu < \eta\).

If a heat exchanger is made with heat pumps, then the efficiency of the installation can be significantly higher than 100% (8), but the degree of perfection of the installation can become equal to 100% only if all losses are actually absent.

5

In most modern thermal installations the primary source of energy is the chemical energy of the fuel burned in the installation. Thus, in a power installation we have the following scheme of energy transformation (Fig. 3).

Strictly speaking, in this case the name “heat engine” should be applied to the energy processes \(B—C\) of the scheme. If this point of view is adopted, then relations (8), (9), and (10), given above, are fully applicable here as well. But in reality it is necessary to take into account the entire chain of energy transformations \(A—B—C\).

The amount of heat released during fuel combustion is determined by the calorific value of the fuel. The numerical value of the latter usually changes only slightly with the conditions of combustion. The qualitative aspect may change much more substantially. Thus, by increasing the preheating of the air (and fuel) at the expense of the heat of the exhaust gases, the mean temperature of heat release during combustion can be increased, which means the possibility of extracting a larger amount of work from each calorie of heat. On the other hand, it is not difficult to imagine a case in which the heat release during fuel combustion will occur at moderate temperatures, differing little from the ambient temperature. This practically deprives us of the possibility of obtaining work from this heat.

Fig. 3

Therefore, when fuel is used as an energy source in a power installation, we make a mistake if we evaluate the fuel by its calorific value. In any case, the calorific value of the fuel does not characterize the quantity \(E_{\max}\)—the maximum work that can be obtained from it under the most favorable conditions. Such conditions will be those of a quasistatic (reversible) combustion reaction, i.e., the combination of the fuel with the oxygen of the air. The magnitude of the maximum work of this reaction must be calculated according to the rules of physical chemistry [9], [10].

Thus, the \(C\) of a power installation operating at the expense of fuel must be calculated from the relations

\[ \mu=\frac{AL}{A_m}, \tag{14} \]

\[ \mu=\frac{860\,N}{BA_m}, \tag{15} \]

where \(A_m\) is the maximum work of the fuel-combustion reaction.

Similarly, for a heating installation operating at the expense of the expenditure of fuel, we obtain, proceeding from (11),

\[ \mu=\frac{u_2 A l_2}{u_m A l_2}. \tag{a} \]

Here, as before,

\[ u_2 A l = Q_2-u_2 T_0 \Delta s_2. \tag{b} \]

In an ideal installation, the work expended to drive the heat pump must be equal to the maximum work of the fuel-combustion reaction

\[ u_m A l_2=BA_m. \tag{c} \]

Consequently, the \(C\) of a heating installation is

\[ \mu=\frac{Q_2}{BA_m}\left(1-\frac{T_0}{T''_{cm}}\right). \tag{16} \]

The question of calculating the maximum work of the fuel-combustion reaction \(A_m\) is considered below.

6

Relations (15) and (16) represent an application of the general equation

\[ \mu=\frac{E_{\mathrm{act}}}{E_{\max}} \tag{5} \]

to power and heating installations operating at the expense of the expenditure of fuel. Their analysis makes it possible to compile analogous relations for a combined (heat-and-power) installation.

If a heat-and-power installation consumes per hour \(B\) kg of fuel with maximum work of the combustion reaction \(A_m\ \dfrac{\text{kcal}}{\text{kg}}\), then

\[ E_{\max}=BA_m\,\frac{\text{kcal}}{\text{h}} \tag{a} \]

of work. The utilized energy consists of \(860N\ \dfrac{\text{kcal}}{\text{h}}\) of work and \(Q_u\ \dfrac{\text{kcal}}{\text{h}}\) of heat at the mean temperature of the heat carrier \(T_{cm}\). According to (16), this is equivalent to the utilization of

\[ Q_u\left(1-\frac{T_0}{T_{cm}}\right)\frac{\text{kcal}}{\text{h}} \]

of work that is necessary in order to obtain this heat by the most economical path (reversible processes). Therefore

\[ E_{\text{use}} = 860N + Q_u \left(1 - \frac{T_0}{T_{cm}}\right), \tag{в} \]

and \(C\) of the thermal power plant

\[ C = \frac{860N + Q_u \left(1 - \frac{T_0}{T_{cm}}\right)}{BA_m} - 1) \tag{17} \]

If the heat installation delivers heat of several parameters

\[ \left(Q_u \frac{\text{kcal}}{\text{h}} \text{ at } T_{cm}; \quad Q'_u \frac{\text{kcal}}{\text{h}} \text{ at } T'_{cm} \text{, etc.}\right), \]
then its \(C\)

\[ C = \frac{860N + \sum Q_u \left(1 - \frac{T_0}{T_{cm}}\right)}{BA_m} - 1) \tag{18} \]

The maximum work of the fuel-combustion reaction, with respect to which the value \(C\) of the thermal power plant is determined, is calculated on the basis of the usual methods of physical chemistry. For gaseous fuel, which may be regarded as a mixture of gases, the following relation may be obtained for calculating the maximum work:

\[ \begin{aligned} A_m ={}& a_{co}CO + a_{h_2}H_2 + a_{ch_4}CH_4 + a_{c_2h_2}C_2H_2 + a_{c_2h_4}C_2H_4 + a_{\Delta V}\Delta V_{on} \\ &- 0.204\,T \lg \frac{(CO_2 \cdot N_2 \cdot H_2O')V_c^{-\Delta V_{on}} \left(\frac{p_0}{p}\right)^{[1+V_u-(V_c-\Delta V_{on})]}} {CO_2 \cdot CO \cdot \ldots \cdot N_2 \cdot H_2O \cdot 0.598V_u} \end{aligned} \tag{19} \]

\[ \underline{\phantom{aaaaaaaaaaaaaaaa}} \]

*) The energy derivation of the value \(\mu\) may be replaced by a “thermal” one. By expending the work \(BA_m \, \frac{\text{kcal}}{\text{h}}\), it is possible, by reversible processes, to obtain

\[ E'_{\max} = \frac{BA_m}{1 - \dfrac{T_0}{T_{cm}}} \quad \frac{\text{kcal}}{\text{h}} \]

of heat at MTT \(T_{cm}\). The energy used should be taken as \(Q_u \, \frac{\text{kcal}}{\text{h}}\) at the same MTT and \(860N \, \frac{\text{kcal}}{\text{h}}\) of work, by means of which one can obtain, at MTT \(T_{cm}\),

\[ \frac{860N}{1 - \dfrac{T_0}{T_{cm}}} \quad \frac{\text{kcal}}{\text{h}} \]

of heat. Hence

\[ E'_{\text{use}} = \frac{860N}{1 - \dfrac{T_0}{T_{cm}}} + Q_u. \]

Substitution into (5) gives the former result.

Here \(CO, H_2 \ldots N_2, H_2O\) are the percentage composition by volume of the combustible gas; \(CO'_2, N'_2, H_2O'\) are the percentage composition of the products of complete combustion when burned in the theoretically required quantity of air; \(V_0\), \(\frac{\mathrm{Nm}^3}{\mathrm{Nm}^3}\), by volume;

\(V_z\) is the quantity of combustion products per \(1\ \mathrm{Nm}^3\) of gas, in \(\mathrm{Nm}^3\), when it is burned in the theoretically required quantity of air;

\(\Delta V_{vp}\) is the quantity of water vapor condensed per cubic meter of gas (fuel), in \(\mathrm{Nm}^3\).

The quantities \(V_0\), \(V_z\), and \(\Delta V_{vp}\) can be calculated from the following relations [11]:

\[ V_0 = \frac{0.50CO + 2.00CH_4 + 2.50C_2H_2 + 3.00C_2H_4 + 0.5H_2 - O_2}{21}, \tag{20} \]

\[ V_z = 0.79V_0 + \frac{CO_2 + CO + 3.00CH_4 + 3.00C_2H_2 + 4.00C_2H_4 + H_2 + H_2O}{100}, \tag{21} \]

\[ \Delta V_{vp} = \frac{0.01\left(H_2 + 2.00CH_4 + C_2H_2 + 2.00C_2H_4 + H_2O\right) - \frac{p'}{p}V_z} {1 - \frac{p'}{p}}. \tag{22} \]

In the last formula, \(p\) is the actual elasticity of the surrounding medium, and \(p'\) is the partial pressure of the water vapor saturating the space at the ambient temperature \(T\). If the calculation gives

\[ \Delta V < 0, \]

then this will mean that condensation of water vapor in the gases, when they are cooled to the ambient temperature, will not occur; i.e., one must take

\[ \Delta V_{vp} = 0, \]

and

\[ H_2O' = -\frac{p'}{p}\cdot 100, \tag{23} \]

\(a_{co}, a_{H_2}\ldots a_{C_2H_4}\) represent the maximum works of the combustion reaction of one hundredth of a normal cubic meter of \(CO, H_2,\ldots C_2H_4\) at an elasticity \(p_0 = 760\ \mathrm{mm\ Hg}\) and ambient temperature \(T\).

\(a_{\Delta V}\) is the maximum work of condensation of one hundredth of a normal cubic meter of \(H_2O\) at \(p_0 = 760\ \mathrm{mm\ Hg}\) and \(T\).

To calculate the quantities \(a_{co}, a_{H_2}\ldots a_{C_2H_2}, a_{\Delta V}\), standard tables of maximum works [12], [13] and values of heat capacities obtained from spectrographic data [14] were used. The results of the calculations are given in Table 1.

Table 1

\(t^\circ C\)	\(a_{co}\)	\(a_{H_2}\)	\(a_{CH_4}\)	\(a_{C_2H_4}\)	\(a_{C_2H_2}\)	\(a_{\Delta V}\)
\(-50\)	28.21	24.73	85.34	138.56	132.05	200
\(-25\)	28.00	24.62	85.31	138.46	131.73	162
\(0\)	27.79	24.49	85.29	138.40	131.41	124
\(+25\)	27.53	24.37	85.26	138.33	131.10	92
\(50\)	27.28	24.25	85.22	138.26	130.79	64

Within the limits of \(\pm 50^\circ\mathrm{C}\), the data of this table fit well the empirical formula

\[ a=A-Bt \tag{24} \]

Table 2

Gas	\(A\)	\(B\cdot 10^{4}\)	Note
CO	27.79	93
H\(_2\)	24.48	48
CH\(_4\)	85.29	12
C\(_2\)H\(_4\)	138.40	30
C\(_2\)H\(_2\)	131.41	127
\(\Delta V_{\mathrm{vp}}\)	124	12950	from \(0^\circ\mathrm{C}\) to \(+50^\circ\mathrm{C}\)
\(\Delta V_{\mathrm{vp}}\)	124	14380	from \(0^\circ\mathrm{C}\) to \(-50^\circ\mathrm{C}\)

In the last term of relation (19), in order to shorten the notation, special designations have been introduced:

\[ CO_{2}= \left(\frac{CO_{2}}{100}\right)^{\frac{CO_{2}}{100}}, \]

\[ CO\cdot = \left(\frac{CO}{100}\right)^{\frac{CO}{100}}, \]

\[ CO_{2}^{1}= \left(\frac{CO_{2}^{1}}{100}\right)^{\frac{CO_{2}^{1}}{100}}, \]

\[ N_{2}^{1}= \left(\frac{N_{2}^{1}}{100}\right)^{\frac{N_{2}^{1}}{100}}, \]

\[ H_{2}O^{1}= \left(\frac{H_{2}O^{1}}{100}\right)^{\frac{H_{2}O^{1}}{100}}. \]

The calculation of these quantities can be facilitated by using the graph (Fig. 4). The first five terms of the relation for the maximum work (19) represent the maximum work of the combustion reaction of \(nm^{3}\) of combustible gas, taken at the elasticity \(p_{0}=760\) mm of mercury and temperature \(T\). By analogy with the corresponding value of the calorific capacity, it may conventionally be called the lower limit of the maximum work. The sixth term of expression (19) gives an addition to the maximum work caused by the condensation of water vapor, which corresponds to the addition in passing to the higher limit of the calorific capacity. The last term of relation (19) is a specific addition to the work caused by the dependence of the maximum work on elasticity. It also takes into account the inequality of the partial pressures of the separate initial and final gases to the total pressure, and the difference between the total pressure of the surrounding medium \(p\) and the standard \(p_{0}\).

Fig. 4

The accuracy of calculating the maximum work of the combustion reaction of gases by (19) depends on the accuracy with which the composition of the combustible gas, \(CO_2, CO,\ldots\), is determined. The use for this purpose of Orsat-type instruments commonly employed in engineering does not provide sufficient accuracy for such a determination. Therefore, despite the theoretical correctness of relation (19), it is preferable to determine the maximum work of the gas-combustion reaction from the relation

\[ A_m = H - T \Delta S, \tag{25} \]

where \(H\) is the heat effect (calorific value), and \(\Delta S\) is the decrease in the absolute value of the entropy in the combustion reaction. Calculation of the latter gives

\[ \Delta S = s_{co}CO+s_{H_2}H_2+s_{CH_4}CH_4+s_{C_2H_2}C_2H_2+s_{C_2H_4}C_2H_4+s_{\Delta V}\Delta V_{vp}+ \]

\[ +0.204\lg \frac{ (CO_2' : N_2' : H_2O')\, \left[ V_2-\ldots V_{vp}\left(\frac{p_0}{p}\right)^{\{1+V_0-(V_2-\ldots V_{vp})\}} \right] }{ CO_2\cdot CO\cdot H_2\ldots N_2\cdot H_2O\cdot 0.598\,V_0 } \tag{26} \]

Here, in addition to the designations given above, \(s_{co}, s_{H_2}\ldots\) represent the decrease in the absolute value of the entropy of \(100\,nm^3\) of \(CO, H_2\ldots\) during their combustion at pressure \(p_0=760\) mm Hg and temperature \(T\). \(s_{\Delta v}\) gives the decrease in entropy during condensation of \(H_2O\). The results of calculations of these quantities are given in Table 2a.

Table 2a

\(t^\circ C\)	\(s_{co}\)	\(s_{H_2}\)	\(s_{CH_4}\)	\(s_{C_2H_2}\)	\(s_{C_2H_4}\)	\(s_{\Delta V}\)
\(-50\)	0.00911	0.00459	0.00161	0.00315	0.01112	1.552 (ice)
\(-25\)	0.00952	0.00470	0.00143	0.00298	0.01121	1.540
\(0\)	0.00959	0.00479	0.00129	0.00284	0.01128	1.538
\(0\)						1.294 (water)
\(25\)	0.00964	0.00487	0.00116	0.00271	0.01134	1.262
\(50\)	0.00968	0.00494	0.00105	0.00259	0.01138	1.229

Within the limits of \(\pm 50^\circ C\), the data of the table are well represented by the empirical formula

\[ s=A_1+B_1t \tag{27} \]

Table 3

GAS	\(A_1\cdot 10^5\)	\(B_1 10^7\)	Note
\(CO\)	959	27
\(H_2\)	478	35
\(CH_4\)	129	\(-56\)
\(C_2H_4\)	284	\(-56\)
\(C_2H_2\)	1128	26
\(V_{vp}\)	153800	\(-990\)	from \(0^\circ C\) to \(-50^\circ C\)
\(V_{vp}\)	129900	\(-140\)	from \(0^\circ C\) to \(+50^\circ C\)

Calculation of the values of the calorific value and the minimum works of the combustion reaction for a number of combustible gases (\(H\) from 700 to \(8000\,\frac{\text{kcal}}{nm^3}\)) showed that these quantities differ from one another by 8–11% (maximum). Therefore, although the calculation of \(A_m\) by (25) and (26) also requires determining the composition of the combustible gas, less stringent requirements may be imposed on this determination.

requirements in the sense of accuracy, since the value of the calorific capacity of the gas will be of primary importance. These same calculations also showed that the magnitude of the last term of relation (19) only in exceptional cases exceeds 1% of the magnitude of the maximum work, and usually lies within 0.2--0.7% of it. If, however, this

Fig. 5

quantity is neglected, then the remaining terms will give the upper limit of the maximum work, and for

\[ \Delta V_{an}=0 \]

we obtain the value of the lower limit of the same quantity.

A comparison of the values calculated in this way for the lower limit of the maximum work of the combustion reaction \(A_{mn}\) and the lower limit of the calorific capacity \(H_n\) showed that they can be connected by the relation

\[ A_{mn}=\left(\frac{100-\Delta_n}{100}\right)H_n, \tag{23} \]

where, with very high accuracy (error 0.25--1.5% maximum),

\[ \Delta_n=\frac{0.27}{0.3125}(250\pm t)^{0}/_{0}, \tag{29} \]

for \(H_n\)

\[ H_n<4000\ \frac{\mathrm{kcal}}{\mathrm{Nm}^{3}} \]

and

\[ \Delta_n=\frac{10^{12}}{3.80}(250+t)^{0}/_{0} \tag{30} \]

for \(H_n\)

\[ H_n>4000\ \frac{\mathrm{kcal}}{\mathrm{Nm}^{3}}. \]

The value of \(\Delta_n\) can be calculated from the graph (Fig. 5).

The calculation of the upper limit of the maximum work (which, neglecting the last term in (19) and (26), gives the actual value of the maximum work of the gas-combustion reaction) can thereafter be made from the relation

\[ A_t = A_{t k} + \nabla_v, \tag{31} \]

where

\[ \nabla_v = a_{\Delta v}\,\Delta V_{v t} \tag{32} \]

The calculation of \(\Delta V_{v t}\) and \(a_{\Delta v}\) was discussed above. \(\Delta v\) may also be read from the nomogram (Fig. 6).

Thus, we have the following methods for determining the maximum work of the combustion reaction of a gaseous fuel:

1. Exact calculation from relations (19) or (25) with (26). In the latter case, in addition to knowing the composition of the combustible gas, determination of its calorific value is also required. The accuracy of determining the maximum work in this case practically coincides with the accuracy of determining the calorific value. The accuracy of calculation by (19) is determined by the accuracy of the gas analysis.

Fig. 6

2. Neglecting the last term in (19) and (26) in calculations of \(A_t\) gives an inaccuracy of no more than 1%, while simplifying the computational procedure.

3. Calculation by relations (28)--(31) gives the value of the maximum work of the combustion reaction of gases with an error of no more than 1% for a calorific value

\[ H_n > 4000 \frac{\text{kcal}}{\text{nm}^3} \]

and 2.5% for fuel with

\[ H_n < 4000 \frac{\text{kcal}}{\text{nm}^3}. \]

4. Taking into account the actual value of \(A_t\) and \(H\), one may assume, with an error of up to \(\pm 5\%\),

\[ A_t = 0.95H. \tag{33} \]

Calculation of the maximum work of the combustion reaction of a solid fuel by a relation of type (19) is practically impossible, since usually only its elemental composition is known. It is possible to use the relation

\[ A_m = \bar H - T \Delta S, \tag{25} \]

if it is possible to calculate the decrease of entropy during combustion, \(\Delta S\). For the latter one may write the relation

\[ \Delta S = S_m + S_{V_0} - S_{V_z} + S_{\Delta v}\Delta V_{vn} + \]

\[ +\,0.204 \lg \frac{ (RO_2 \cdot N_2 \cdot H_2O)^{V_z-\Delta V_{vn}} \left(\dfrac{P_o}{P}\right)^{[V_0-(V_z-\Delta V_{vn})]} }{ 0.598^{V_0} }, \tag{34} \]

where \(S_m\), \(S_{V_0}\), \(S_{V_z}\) are the absolute values of the entropies of the fuel, of the theoretically required amount of air for its combustion, and of the resulting combustion products per 1 kg of fuel; \(RO_2\), \(N_2\), and \(H_2O\) are the percentage composition of the combustion products.

As before,

\[ RO_2'=\left(\frac{RO_2}{100}\right)^{\frac{RO_2}{100}} \]

and so forth.

All quantities on the right-hand side, except for the value of the entropy of the fuel \(S_m\), are amenable to calculation. To calculate \(S_m\) it is necessary to know the heat capacity of the fuel down to low temperatures. Since these data are not yet available to us, we are forced to resort to approximate calculations.

To draw up calculation formulas, the known approximate Nernst formulas are used for the sublimation process at atmospheric pressure,

\[ S_z - S_m = 7.4 \lg T + 14\frac{ze}{\text{mole}}. \tag{35} \]

If the combustion process is imagined as a reaction between solids with subsequent sublimation of the combustion products, then the change of entropy at normal pressure \(p_o\) and temperature \(T\) will be

\[ \Delta S_o = (S_m-\Sigma S'_m) + (V_o - V_z)(0.625 + 0.330 \lg T), \tag{36} \]

where \(\Sigma S'_m\) is the sum of the entropies of the reaction products in solid form. The entire first bracket of the right-hand side of (36) gives the entropy change in the reaction between solids. For Nernst bodies at \(T=0\) this quantity is equal to zero. It may be assumed that at ordinary ambient temperatures up to \(t=+50^\circ C\) this quantity also differs little from zero. If, moreover, it is taken into account that at \(t>0\) the moisture of the fuel \(W^p\) may be regarded as being in the liquid state rather than as ice, and also the possibility of condensation of part of the water vapor from the combustion products \(\Delta V_{vn}\), then the decrease of entropy during combustion will be

\[ \Delta S = (V_o - V_z)(0.625 + 0.330 \lg T) + S_{\Delta v}\Delta V_{vn} + 0.293\frac{W^p}{100} + \]

\[ +\,0.204 \lg \frac{ (RO_2 N_2 H_2O \cdot \gamma)^{V_z-\Delta V_{vn}} \left(\dfrac{P_o}{P}\right)^{[V_0-(V_z-\Delta V_{vn})]} }{ 0.598^{V_0} }. \tag{37} \]

At \(t<0\), the moisture of the fuel must be considered to be in the form of ice; then the term

\[ 0.293\frac{W^p}{100} \]

should be omitted from the equation.

The admissibility of calculating the quantity \(\Delta S\) from relation (37) was verified by comparing calculations made with it against calculations by formula (34) for a number of compounds for which entropy values are available in standard tables. In all cases the difference in the magnitude of the maximum work proved insignificant. It is determined mainly by the oxygen content. With a weight content of

\[ O_2 > 45\%, \]

this difference reaches up to 1.5%, while at a lower oxygen content it is less than 0.5%.

On the other hand, the use of relations (25) and (37) for calculating the maximum work of the combustion reaction of a number of solid fuels showed that taking into account the last term of (37) leads to a refinement of the value of \(A_m\) by no more than 0.5%. Therefore it is quite possible to omit it, which considerably simplifies the calculation technique.

Thus, on the whole, it may be considered that the use of relations (25) and (37) (without taking account of the last term) for calculating the maximum work of the combustion reaction of solid fuel, \(A_m\), makes it possible to obtain this quantity with an error of up to 1% for hard coals and anthracites and not more than 2% for young chemical fuels (brown coal, peat, firewood). The difference between the value of the maximum work \(A_m\) and the calorific value \(H\) does not exceed 2.5–3% for young chemical fuels and is less than 2% for hard coals and anthracites.

Fig. 7

A practically very convenient method for calculating the maximum work of the combustion reaction of solid fuel is the use of the nomogram in Fig. 7. It makes possible the calculation of the quantity

\[ \Delta_{\mathrm{н}} = \left[ (1.429 + 0.00052t)(V_2 - V_o) - 0.293\,\frac{W^p}{100} \right], \tag{38} \]

which enters into the relation for the lower limit of the maximum work,

\[ A_{m\mathrm{н}} = H_{\mathrm{н}} + \Delta_{\mathrm{н}} . \tag{39} \]

In this case, for \(t < 0\), the last term in brackets in (38) should be omitted, i.e., one should take

\[ W^p = 0, \]

After this, the true value \(A_t\) is calculated from (31) or from the nomogram of Fig. 6.

The calculation shows that the value \(A_t\) changes little within the usual intervals of ambient temperature \((0\) to \(+50^\circ\mathrm{C}\) and \(0\) to \(-50^\circ\mathrm{C})\). When passing through \(0^\circ\mathrm{C}\) there is a sharper change, due to the different state of aggregation of water.

9

In principle, for calculating the maximum work of the combustion reaction of liquid fuel, relations (25) and (34) are suitable. In practice, here too we are forced to take the path of approximate relations.

Reasoning analogous to that used in deriving (36) leads to the relation

\[ \Delta S = (S_m - \Sigma S'_m) + \Delta S_{pl} + (V_o - V_z)(0.625 + 0.330\lg T), \tag{40} \]

where \(\Delta S_{pl}\) is the increase in entropy upon melting. But verification of this relation for hydrocarbons of various series showed that the results do not agree sufficiently well with reality. Moreover, this method is not very convenient, because usually we do not have sufficiently complete data on the quantitative content of different hydrocarbons in petroleum. Meanwhile, different hydrocarbons differ substantially in the magnitude of \(\Delta S_{pl}\).

Therefore an attempt was made to select an empirical formula for calculating the change in entropy in the combustion reaction of liquid fuel, which led to the relation

\[ \Delta S = 0.300 + (V_o - V_z)(0.625 + 0.330\lg T) + S\Delta V\,\Delta V_{vn}. \tag{41} \]

Comparison of the values of the maximum work of the combustion reaction according to relations (25), (41), (25), and (34) for various hydrocarbons\(^{1)}\) shows that the discrepancy will be no more than \(40\text{--}50\,\dfrac{\mathrm{kcal}}{\mathrm{kg}}\), which for petroleum gives an error of no more than \(0.5\%\). The admissibility of using (41) was also revealed for other compounds (alcohols, acetone, nitrogen compounds).

In practice, for calculating the maximum work of the combustion reaction of liquid fuel it is convenient to use the nomogram of Fig. 7, where for liquid fuel

\[ \Delta_n = [(V_z - V_o)(1.429 + 0.00052t) - 0.300]\,T \tag{42} \]

and then according to (39) and according to (31), or from the nomogram of Fig. 6.

10

Comparison of the relations for the efficiency and \(C\) of a purely power installation operating by combustion of fuel shows that their difference is due to the difference between the heating value \(Q^p\) and the maximum work of the fuel-combustion reaction \(A_t\). As indicated, this difference is small: a maximum of \(10\%\) for gaseous fuels and considerably less for solid and liquid fuels. Therefore, with respect to a power installation of this type

\(^{1)}\) The check was carried out for 25 hydrocarbons of the methane series, 7 hydrocarbons of the naphthene series, and 13 aromatics. For calculations according to (34), data of Parks and Huffman [15] were used.

$C$, estimated by the value of $\eta$, practically corresponds to reality.

In estimating the degree of perfection of a heating installation the situation changes substantially. With numerically close values of $Q^p$ and $A_m$, the calculation of $C$ by (16) gives values sharply different from the value of the efficiency of this installation. In Fig. 8 are given the results of calculations of $C$ for two heating installations: a water-heating one, with a hot-water temperature of $95^\circ\mathrm{C}$ and a return temperature of $70^\circ\mathrm{C}$, and a steam one with steam pressure $p = 1.2$ ata. The efficiency of the installation is taken as 60 and 75% for the water and steam installations, respectively. As is seen from the figure, the degree of perfection of the heating installation is very small (13–27%). In any case, $C$ of the installation is considerably less than its efficiency. Further, the value $C$ of the heating installation depends appreciably on the temperature of the surrounding medium (more precisely, on the ratio $\dfrac{T_0}{T_1}$), decreasing as it increases.

Comparison of the numerical values of the efficiency and $C$ shows clearly that the latter indeed correctly estimate the degree of perfection of the given installation. High and unchanging values of the efficiency of the installation indicate only that, of the thermal effect of the fuel-combustion reaction, the same fraction of heat is used in the heating steam or heating water.

Fig. 8                Fig. 9

But if, for example, the mean temperature of the heating agent were equal to the temperature of the surrounding medium $t_0$, then this heat could be obtained from the surrounding medium “free of charge,” and consequently there would be no grounds for considering the degree of perfection of such an installation higher than 0 at any value of its efficiency.

A particularly sharp essential difference in the estimate by efficiency (1) and by $C$ (18) is obtained for a combined heat-and-power installation. In Fig. 9 are given the results of calculations for a back-pressure installation as a function of the initial steam pressure. In all calculations it was assumed:

\[ t_1 = 450^\circ\mathrm{C},\quad p_2 = 1.2\ \text{ata},\quad t = 80^\circ\mathrm{C},\quad t_{\text{c.w.}} = 40^\circ\mathrm{C},\quad \eta_{\text{bo}} = 0.80, \]

\[ \eta_M = 0.98,\quad \eta_i = 0.96,\quad t_0 = -20^\circ\mathrm{C}. \]

For curve No. 1 it was taken: $x = 100\%$, $\kappa = 100\%$,

“  ” No. 2 “ $x = 100\%$, $\kappa = 75\%$,

“  ” No. 3 “ $x = 70\%$, $\kappa = 100\%$,

“  ” No. 4 “ $x = 70\%$, $\kappa = 75\%$,

where $x$ is the percentage of utilization of exhaust steam and
$\kappa$ is the percentage of condensate return.

In Fig. 10 are shown the results of analogous calculations at unchanged boiler pressure $p = 60$ ata.

Examination of these graphs leads to the conclusion that an assessment of the degree of perfection of a thermal-power installation by efficiency gives clearly inflated results. The influence of changes in the steam parameters affects the economy of a back-pressure installation much more sharply than its efficiency. Thus, according to curve 4 of Fig. 8, an increase in the initial pressure from 18 to 120 ata increases the efficiency of the installation by 2% of the initial value of 59.5%, whereas the increase in \(C\) (\(\mu\)) amounts to 5.4% of the initial value of 27.5%. An increase in steam superheat (curves 4 of Fig. 9) from 350°C to 500°C raises \(\eta\) from 60.4 to 61.1%, and the value of \(\mu\) from 30.5 to 31.8%. As is known, experience confirms the fact that increasing the initial pressures and temperatures of the steam in a combined installation is advantageous.

Fig. 10

A simple solution is also found for the well-known paradox of the “advantage” in a back-pressure installation of worsening the process of energy generation (an increase in \(\eta\) when passing to throttling of the steam). Calculations show that when the process of energy generation is worsened, \(C\) of the installation (\(\mu\)) decreases, despite the increase in its efficiency \((\eta)\).

Finally, the fundamental relation (5) may be applied instead of (2) to assess the degree of perfection of the individual stages of energy transformation. In Table 4 below, a comparison is given of the values of \(\eta\) and \(\mu\) for a steam-power installation with the following data:

Fuel—hard coal, Anzhero-Sudzhensk; air preheating—200°C, \(q_3 = 0\), \(q_4 = 4\%\), \(q_5^m = 0.5\%\), \(q_5^k = 1\%\). Steam pressure \(p = 32\) ata, \(t = 425^\circ\mathrm{C}\). Before the turbine \(p = 29\) ata, \(t = 400^\circ\mathrm{C}\), \(\eta_{oi} = 0.80\), \(\eta_m = 0.98\), \(\eta_g = 0.96\).

Table 4

$\eta_m$	$\eta_{ku}$	$\eta_{pp}$	$\eta_t$	$\eta_u$
95.5	85.0	98.4	37.0	23.2
$\mu_g$	$\mu_{ku}$	$\mu_{pp}$	$\mu_M^0$	$\mu_u$
71.7	33.7	96.6	95.2	23.2

Here the notation is:

$\mu_g$ — $C$ of the furnace process,
$\mu_{ku}$ — $C$ of the boiler installation,
$\mu_{pp}$ — $C$ of the steam line,
$\mu_M^0$ — theoretical $C$ of the machine,
$\mu_u$ — $C$ of the installation.

A comparison of the tabulated figures shows that, despite the high efficiency of the boiler installation, its degree of perfection is very small. With small, on the whole, furnace losses of 4.5%, even with the most perfect method of utilizing the heat of the combustion products one can obtain only 71.7% of the maximum work of the fuel-combustion reaction. Consequently, the energy losses of the furnace are considerably greater than its direct losses. Especially significant is the qualitative loss of energy in the boiler installation, where the steam obtained has a considerably lower $C$—TT than the combustion products.

The $C$ of the steam line is also lower than its efficiency, which is due to the presence of throttling, lowering the $C$—TT of the steam.

Conversely, an estimate of the degree of perfection of the theoretical cycle of the machine by its thermal efficiency $\eta_t$ gives a number considerably smaller than its $C$ $\mu_M^0$. Indeed, the Rankine cycle of a steam installation, as is known, differs only slightly from the Carnot cycle most advantageous under the given conditions. Therefore there is no basis for evaluating its degree of perfection by the efficiency value, which is inevitably low under the given conditions.

Neglecting, for solid fuel, the difference between the values of the heating value and the maximum work of the combustion reaction, we obtain for a purely power installation

\[ \eta_u \simeq \mu_u, \]

i.e., approximately the same numerical result. But the distribution of losses according to the magnitude of $C$ of the individual parts of the installation indicates that an increase in its overall $C$ is possible to a serious extent through better organization of the operation of the boiler part, and not through improvement of the theoretical process of the machine.

11

The fundamental relation (5) also makes it possible to write a general formula for the $C$ of an energy system. The energy used in the system consists of

a) electrical energy $860\,N_e\,\dfrac{\mathrm{kcal}}{\mathrm{h}}$,

b) other kinds of work, for example, mechanical energy, compressed air, water raised to a certain height, etc., in the amount

\[ 860 \sum N_n \frac{\mathrm{kcal}}{\mathrm{h}}; \]

c) thermal energy \(Q'_u \dfrac{\mathrm{kcal}}{\mathrm{h}},\ Q''_u \dfrac{\mathrm{kcal}}{\mathrm{h}} \ldots\) at \(C-\mathrm{TT}\) \(T'_{cm}, T''_{cm}\)

... respectively. At ambient temperature \(T_0\), the availability of this heat is

\[ \sum Q_u \left(1-\frac{T_0}{T_{cm}}\right) = Q'_u \left(1-\frac{T_0}{T'_{cm}}\right) + Q''_u \left(1-\frac{T_0}{T''_{cm}}\right) +\cdots \]

Thus,

\[ E_{\mathrm{isp}} = 860 N_e + 860 \sum N_n + \sum Q_u \left(1-\frac{T_0}{T_{cm}}\right). \tag{a} \]

The maximum quantity of energy that can be obtained in the system consists of

a) the sum of the maximum works of the combustion reactions of the fuels expended in the system

\[ \sum BA_m = B_1 A_{1m} + B_2 A_{2m} + \cdots, \]

b) the direct input of power (hydraulic, electrical, etc.) in the amount

\[ 860 N_z = 860 N_{z э} + 860 N_{z'} + \cdots), \]

c) the input of heat of various parameters, with the corresponding availability

\[ \sum Q^z_u \left(1-\frac{T_0}{T^z_{cm}}\right). \]

Thus,

\[ E_{\max} = \sum BA_m + 860 \sum N_z + \sum Q^z_u \left(1-\frac{T_0}{T^z_{cm}}\right) \tag{в} \]

According to relation (5), \(C\) of the system will be

\[ \psi_c = \frac{ 860\left(N_e+\sum N_n\right)+\sum Q_u\left(1-\dfrac{T_0}{T_{cm}}\right) }{ \sum BA_m + 860\sum N_z + \sum Q^z_u\left(1-\dfrac{T_0}{T^z_{cm}}\right) }. \tag{43} \]

A parallel calculation of the magnitude of the maximum work of the combustion reaction and the calorific capacity of a number of fuels leads to the conclusion that the difference between them is small. For hard coals the difference does not exceed 2%, for young chemical fuels 2.5–3%, and for liquid fuels the difference does not go beyond 4%. The greatest difference is obtained for gaseous fuels, where it reaches 10%. But for gas \(A_m\) is less than \(H\); therefore, if one takes

\[ A_m = 0.95H, \]

then the error will be no more than \(+5\%\). In general, one may take

\[ A_m = mH, \tag{44} \]

where, with the accuracy indicated above, for solid and liquid fuels

\[ m = 1, \]

and for gaseous fuels

\[ m = 0.95. \]

If, in the calculation formulas for \(C\), (14), (15), (16), (17), (18), and (43), the corresponding values from (44) are substituted, then the value of \(\psi\) will be obtained with the same error. But at the present stage not a single thermal-power installation utilizes even 50% of the maximum work of the fuel-combustion reaction. Further, in estimating \(C\) for thermal-power installations, we are usually interested not in absolute but in relative quantities. Therefore, as a first approximation, the calculation of \(C\) for thermal-power installations with the indicated replacement may be considered admissible. The value obtained, so as not to confuse it with the true value \(C\), may be called “thermal \(C\)” and denoted by \(\mu_t\). Thus, for a thermal-power installation, instead of (18) we have, for thermal \(C\),

\[ \mu_t = \frac{860N + \sum Q_u \left(1 - \frac{T_0}{T_{cm}}\right)} {mBQ^p} \tag{45} \]

and so on.

References Cited

1. I. N. Butakov—General expression of efficiency for various thermal-power installations. Teplosilovoe khozyaistvo, 1938.

2. V. Yakub—Thermal electric stations, 1935.

3. I. Gofman—Organization and planning of the power industry, 1939.

4. M. Khazen—A new efficiency system for steam-turbine installations, 1935.

5. M. Khazen—A new method for estimating the degree of perfection of a thermal-power installation. Heat and Power, 1936.

6. M. Yanovskii—On a new method for estimating the perfection of a steam-power installation. Soviet Boiler and Turbine Construction, 1937.

7. G. Fuks—Mean thermodynamic temperatures. Proceedings of the Tomsk Polytechnic Institute, vol. 63, 1944.

8. Pakshver—(Paper from BBC Mitteilungen). A thermocompressor with mechanical drive in industrial installations. Electric Stations, 1944.

9. A. Rakovskii—Physical Chemistry, 1939.

10. G. Fuks—Thermodynamic estimation of the degree of perfection of thermal-power installations. Tomsk, 1941.

11. G. Ulich—Chemical Thermodynamics, 1933.

12. Landolt–Bornstein. Physikalisch-Chemische Tabellen.

13. State Engineering Experimental Station. New specific heat equation. Mechanical Engineering, 1939.

14. Lewis and Randall—Chemical Thermodynamics, 1935.

15. Parks and Huffman—Free Energies of Organic Compounds, 1936.