EFFICIENCY OF A HEAT-POWER PLANT AND AN ENERGY SYSTEM

I. N. Butakov

“Only bold and open criticism helps our people to improve themselves, impels them to move forward, to overcome shortcomings in their work. Where there is no criticism, there stagnation and inertia take root; there is no place for forward movement.”

A. A. Zhdanov

Introduction

Wishing to obtain an idea of the merit of any machine, one determines the ratio of the useful work produced by it to the work of the motive forces, taking, as far as possible, a long interval of time, and calls this ratio the efficiency of the machine. In determining the efficiency for heat engines, we understand by it the ratio of the quantity of heat actually converted into work to the quantity of heat expended (supplied). As is known, in heat-power installations efficiencies may have different values depending on the stage of the thermal process being considered. Great importance is attached to the absolute, or economic, efficiency, representing the ratio of the quantity of heat used in the form of work to the heat of the fuel which the latter could have given theoretically if all losses had been avoided. In our time, useful work is usually obtained in the form of electrical energy at the station busbars, so that the economic efficiency of a purely condensing station, when generating \(E\) kWh over a definite interval of time, may be represented in the form

\[ \eta_k = \frac{860 \cdot E}{Q_{top}}, \]

where \(Q_{top}\) is the heat contained in the fuel consumed under the station boilers during the interval of time considered.

Karl Marx, on the first page of Volume 1 of Capital, pointed out that a commodity, as a useful thing which by its properties satisfies some human need, is an aggregate of many properties and therefore can be useful by its various aspects. To discover these various aspects, and consequently the manifold ways of using things, is the work of historical development. These thoughts are, more than anywhere else, valid with respect to electric power stations, where the heat of steam for decades in the past produced the sole commodity—electric energy. At that time the heat carried away into the condenser or discharged into the atmosphere proved to be waste, lost. With the appearance of combined heat and power plants, electric stations that had previously produced, as indicated, the sole commodity—electric energy—began to give a new commodity in the form of heat.

for various heating purposes, and the source of this heat is steam that has already previously performed work in prime movers. CHP plants are a sign of our time; they became widespread only in the twentieth century.

The socialist organization of our country has made this principle of the combined use of heat predominant in the energy economy of the USSR, chiefly as a result of the considerable advantages in heat utilization associated with it.

In accordance with the twofold kind of commodities now produced at CHP plants, satisfying equally important human needs, the concept of the absolute efficiency of a CHP plant, in accordance with what has been said above, must be modified, since in the case of complete utilization of the heat of the exhaust steam, in the numerator of the expression for the efficiency of a CHP plant we shall now have two terms: 1) the heat converted into electric power, \(860\cdot E_m\), and 2) the heat of the exhaust steam intended for heating purposes; that is, for the efficiency of a CHP plant we shall have the expression

\[ \frac{860\cdot E_m+W}{Q_{mon}}, \tag{1} \]

where \(W\) is the heat produced at the CHP plant for heat consumers during the time interval under consideration.

Critique of the Critics

Prof. M. M. Khazen [1], analyzing an analogous formula for a back-pressure steam-turbine installation, recognizes it as “meaningless.” His line of reasoning is as follows. If \(G_n\) is the hourly steam consumption in kg for the back-pressure turbine, \(i_1\) is the heat content of the steam before the turbine, \(i'_n\) is the final heat content in adiabatic expansion, \(\eta_i\) is the internal relative efficiency of the turbine, then the heat converted into work is

\[ G_n(i_1-i'_n)\eta_i\cdot\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}}, \]

and the heat delivered for heating purposes is

\[ G_n\left[i_1-(i_1-i'_n)\eta_i\right]. \]

Therefore the coefficient characterizing the losses in the engine will be

\[ 1-\eta_{\mathrm{ek}} = 1- \frac{ G_n(i_1-i'_n)\eta_i\cdot\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}} + G_n\left[i_1-(i_1-i'_n)\eta_i\right] }{ G_n i_1 } = \]

\[ = \frac{(i_1-i'_n)\eta_i(1-\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}})}{i_1}, \tag{2} \]

where \(\eta_{\mathrm{ek}}\) is the economic efficiency of the turbine installation under consideration. From the last expression Prof. Khazen concludes that the losses in an installation with back pressure are the greater, the greater the internal relative efficiency \(\eta_i\) of the turbine; that the losses are equal to zero when there is a complete absence of generation of electric energy (\(\eta_i=0\)). “This is obviously meaningless,” concludes Prof. Khazen.

From expression (2) we have

\[ \eta_{\mathrm{ek}} = 1- \frac{(i_1-i'_n)\eta_i(1-\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}})}{i_1}. \]

But this efficiency is more correctly given another interpretation, taking \(t_{\mathrm{cond}}\) to be the temperature of the condensate of the extracted steam:

\[ \eta_{\mathrm{ek}}= \frac{(i_1-i'_n)\eta_i\eta_{\mathrm{mech}}\eta_{\mathrm{gen}}+i_1-(i_1-i'_n)\eta_i-t_{\mathrm{cond}}}{i_1-t_{\mathrm{cond}}} = \]

\[ = \frac{i_1-t_{\mathrm{cond}}-(i_1-i'_n)(1-\eta_{\mathrm{mech}}\eta_{\mathrm{gen}})\eta_i}{i_1-t_{\mathrm{cond}}} = 1-(1-\eta_{\mathrm{mech}}\eta_{\mathrm{gen}})\frac{h_a\eta_i}{i_1-t_{\mathrm{cond}}} = \]

\[ =\eta'_{\mathrm{mech}}\eta'_{\mathrm{gen}}, \tag{3} \]

so that

\[ (1-\eta'_{\mathrm{mech}}\eta'_{\mathrm{gen}})(i'_1-t_{\mathrm{cond}}) = (1-\eta_{\mathrm{mech}}\eta_{\mathrm{gen}})h_a\eta_i . \]

It follows from this that: 1) the mechanical and electrical losses in an engine with final use of heat must be referred to the entire amount of heat of 1 kg of steam \((i_1-t_{\mathrm{cond}})\) produced in the given case, and not only to the heat drop \(h_a\eta_i\), as is usually done, and as is, of course, correct with respect to a condensing installation; 2) the mechanical and electrical losses \((1-\eta_{\mathrm{mech}}\eta_{\mathrm{gen}})h_a\eta_i\), in absolute magnitude in calories, increase together with the increase of the heat drop \(h_a\eta_i\), which is natural and characteristic also of a purely condensing engine. At the same time, however, the product \(\eta'_{\mathrm{mech}}\eta'_{\mathrm{gen}}\) will decrease, since, out of the total amount of heat produced by each kg of steam \((i_1-t_{\mathrm{cond}})\), a larger amount will now go to mechanical and electrical losses. Thus the emphasis must be placed on the magnitude of the electrical and mechanical losses, and not at all on \(\eta_i\) and \(h_a\), as in Prof. Khazen, whose reduction will lead only to an increase in the production of condensing kWh, and hence to a deterioration of the overall efficiency of the installation, as will be clearly seen from the subsequent exposition. It is precisely in the absence of a proper formulation of the conclusion from formula (2), which leads the reader into bewilderment, that the root of Prof. M. M. Khazen’s error lies.

Fig. 1

At the power-engineering conference held in June 1940 at the Tomsk Polytechnic Institute, Prof. G. I. Fuks attempted to substantiate the unsuitability of formula (1) by saying that in its numerator “we are explicitly adding quantities of different meaning, such as work and heat, which are identical neither in meaning nor in value.” If we take the Rankine cycle in \(TS\) coordinates (Fig. 1), we can convince ourselves that this assertion is not justified. Indeed, the efficiency of this cycle is

\[ \eta_p= \frac{\text{area }ABCDEA}{\text{area }ABCDEA+\text{area }AEIGA}. \]

The numerator and denominator of this efficiency are universally recognized as homogeneous, despite the fact that the quantity proportional to area \(AEIGA\) denotes heat converted into work, while the quantity proportional to area \(AEIGA\) is heat absorbed by the cold source of heat.

with heat close to the latent heat of vaporization at temperature \(T_2\), or even exactly equal to it in the case where point \(E\) lies on the upper limiting curve. But in formula (2) the quantity \(W\) is also heat, proportional to \(n l \cdot AE/GA\); only it is placed in the numerator and referred not to one but to a large number \(\kappa\) of kg of steam, which is no longer a matter of principle. If to this we add that, under practical conditions, the line \(AE\) may occupy different positions and thus the ratio of the two indicated areas may vary depending on the circumstances, then the interchangeability of the latter becomes clear, and consequently the objection of Prof. G. I. Fuks under discussion may be disregarded all the more since \(n l \cdot AE/GA\) at the same time denotes the work performed by steam of initial temperature \(T_2\) during its evaporation, when the temperature of the cold source \(T_2 = 0\). This is quite clear when point \(E\) lies on the upper limiting curve.

From what has been said it is evident that there are no grounds for abandoning the completely correct formula (1) for the efficiency of a back-pressure plant with final utilization of heat

\[ \eta_m=\frac{860\mathcal{E}_m+W}{Q_{\text{top}}}. \]

Formula for the reciprocal of the efficiency of a thermal power plant and its consequences

Owing to the lack of synchronism in the consumption of heating heat and electric energy, it is known that these asynchronous demands for energy products have to be met by installing engines: 1) back-pressure engines (of worsened vacuum), regulated according to the load of the heat consumers; 2) condensing engines, regulated depending on the demand for electric energy. In our time this problem is most successfully solved, in most cases, by installing engines with intermediate steam extraction, representing a combination in one engine of back-pressure and condensing parts. The total heat consumption in such combined plants will be

\[ Q=Q_k+Q_m, \]

or

\[ \frac{860\mathcal{E}+W}{\eta} = \frac{860(\mathcal{E}-\mathcal{E}_m)}{\eta_k} + \frac{860\cdot \mathcal{E}_m+W}{\eta_m}, \tag{4} \]

where \(\mathcal{E}=\mathcal{E}_m+\mathcal{E}_k\) is the total generation of electric energy in kWh over the time interval under consideration, with \(\mathcal{E}_m\) being the generation of kWh for heat consumption, and \(\eta\) the absolute efficiency of the combined plant. From formula (4) it is evident that the efficiencies \(\eta_k\) and \(\eta_m\) characterize particular processes, depending on the nature of each of them. Hence, after simple transformations,

\[ \frac{1}{\eta} = \frac{1}{860\cdot \mathcal{E}+W} \left[ \frac{860(\mathcal{E}-\mathcal{E}_m)}{\eta_k} + \frac{860\cdot \mathcal{E}_m+W}{\eta_m} \right] = \]

\[ = \frac{1}{\eta_m} \left[ 1+ \frac{ \left(1-\frac{\mathcal{E}_m}{\mathcal{E}}\right) \left(\frac{\eta_m}{\eta_k}-1\right) }{ 1+\frac{W}{860\cdot \mathcal{E}} } \right]. \tag{5} \]

From formula (5) for the efficiency of a thermal power plant it follows that: 1) everything that increases the value of the sum in square brackets decreases the efficiency \(\eta\) of the thermal power plant; 2) an increase

of the district’s specific heat supply, in kcal/kWh, \(W/\mathcal{E}\), increases the efficiency \(\eta\);

3) an increase in the ratio \(\mathcal{E}_m/\mathcal{E}\) and in \(\eta_k\)—the economic efficiency of the condensing part of the installation—raises the efficiency \(\eta\), which can be achieved by increasing the initial steam parameters and lowering the back pressure \(\left(\text{for } \mathcal{E}_m/\mathcal{E}\right)\), and also through the development of regeneration.

Starting from formula (3), we have

\[ \eta_m=\eta_{ky}\cdot \eta_{pot}\cdot \eta'_{mech}\cdot \eta'_{gen}, \]

where \(\eta_{ky}\) is the efficiency of the boiler installation. The efficiency of the flow in transporting heat within the station, \(\eta_{pot}\), may be determined from the following: if, for the time interval under consideration, \(Q_{top}\) is the number of calories contained in the fuel burned, \(Q_{kond}\) is the heat loss in the condenser, \(Q_{pot}\) is the heat loss during its transport within the station, \(Q_{potky}\) is the heat loss in the boiler installation; \(W_k\) is the number of calories released in the form of live steam from the boiler, \(Q_{rast}\) is the heat consumption for boiler firing-up, and \(Q_{khtn}\) is the heat consumption for plant service and technical needs (showers, fuel-oil heating, etc.), then

\[ Q_{pot}=Q_{top}-Q_{potky}-860\mathcal{E}-W_k-Q_{kond}-Q_{pot\,mech\,el}-Q_{rast}-Q_{khtn}. \]

Hence

\[ \eta_{pot}=1-\frac{Q_{pot}}{Q_{top}} = \frac{860\mathcal{E}+Q_{potky}+W+W_k+Q_{kond}+Q_{pot\,mech\,el}+Q_{rast}+Q_{khtn}}{Q_{top}}. \]

Figure 2 gives a nomogram for determining \(\eta\) at \(\eta_m=0.76\)¹). From our formula (5), as a special case, one may derive a formula for the efficiency of a condensing installation with an independent boiler house supplying heat directly for heating purposes. In this case \(\mathcal{E}_m=0\), and our formula (5) takes the form

\[ \frac{1}{\eta_{ok}} = \frac{1}{\eta^{ok}_m} \left[ 1+ \frac{\eta^{ok}_m/\eta_k-1}{W/860\cdot \mathcal{E}+1} \right]. \tag{5-a} \]

Since in this case the fractional factor \(\left(1-\mathcal{E}_m/\mathcal{E}\right)\) becomes unity, it is obvious that the efficiency of the installation will be lower than in the case of a combined heat-and-power plant. This is also understandable, since the production of kWh in the presence of a heating boiler house will take place exclusively in a condensing installation with the lower efficiency \(\eta_k\). The efficiency of the heating part of the installation in this case is

\[ \eta^{ok}_m=\eta'_{ky}\cdot \eta'_{pot}. \]

In the special case when \(\mathcal{E}_m=\mathcal{E}\), i.e., when there is a back-pressure (degraded-vacuum) installation with full utilization of the exhaust heat, formula (5) becomes \(1/\eta=1/\eta_m\). In this case the efficiency of the installation reaches its maximum.

If, however, the power station, while being condensing, has regenerative equipment, then in this case the value of the efficiency \(\eta_k\) is increased as a result of an approach to the efficiency of the Carnot cycle, and the more so the greater the number of points

¹) The nomograms in Figs. 2, 4, 5, 7, 10, 11 were constructed by Eng. V. P. Lebedev. He also carried out the work of constructing the graphs in Figs. 3, 6.

selection of steam, while an increase in \(\eta_k\), as is evident from formula (5), raises the overall efficiency \(\eta\) of the installation.

The interpretation of the advantages of the regenerative process as the generation of electric power for heat consumption, originally given by B. M. Yakub in his essentially erroneous review of A. Ya. Yastrzhembsky’s book Technical Thermodynamics (Teplo i Sila, 1933, No. 8, pp. 38–39), cannot be accepted, in view of the contradiction of this “theory” with reality. The review is built on the well-known paralogism of substituting proof—

Nomogram for determining the efficiency of an installation with combined heat production

\[ \frac{1}{\eta} = \left[ \frac{ \left(1-\frac{[[unclear]]}{[[unclear]]}\right) \left(\frac{1}{\eta_0 \eta_k}-1\right) }{ \frac{W}{860}+1 } \right] \]

Fig. 2

—proof, when the author’s position is declared unworthy of attention solely on the basis that errors have been found in some particulars, without assessing whether these errors undermine the author’s position in essence and as a whole. In A. Ya. Yastrzhembsky’s textbook the theoretical cycle of the regenerative process itself is schematized, giving a stepped expansion line, but not giving the same stepped line of the liquid phase and of the limiting lower curves as a result of the operation of variable quantities of the working substance, which require that along the abscissa axis not the quantity \(S\) be plotted, but \(GS\), where \(G\) is the variable quantity of the working substance. But in elementary textbooks of thermodynamics a schematic representation of the regenerative cycle is often encountered, which, of course, gives discrepan-

... with reality, but this does not affect the correctness of posing the question in essence, concerning the nature of regeneration. B. M. Yakub writes further that the full theoretical efficiency of electric-power generation with heat consumption is equal to unity, and no regeneration can increase it; for this reason there can be no question of any approach to the efficiency of the Carnot cycle. The use of regeneration can increase the generation of electric power from a given quantity of heat, correspondingly reducing the delivery of heat to heat consumers or the loss in the condenser. Here, indeed, lies one of the reasons why Yakub’s “theory” diverges from reality, since, for a given heat consumption, in the case of a regenerative process we shall have an increase in electric-power generation in comparison with a purely condensing installation, whereas, when the same quantity of steam is extracted from intermediate stages for external consumers, electric-power generation will be reduced even in comparison with the case of a purely condensing installation. This difference is especially clear when the condensate of the extracted heating steam is not returned to the thermal power plant.

As is known, the newest high-pressure industrial power installations in the USA and Germany are characterized either by the complete absence of regenerative heating of feedwater, or by its extremely weak development. It would seem that, in the case under consideration of industrial thermal power plants, when there is a large heat consumption for technological needs and a corresponding generation of electric power with heat consumption, abandonment of the use of regeneration is expedient in all cases, since the thermal scheme of the station is simplified, thereby increasing the reliability of its operation, while the condensate can be returned to the feed system at a sufficiently high temperature, which should reduce the effect of regeneration. Nevertheless, it must be said that there can be no general solution here, since regeneration, at one and the same fuel consumption, by giving an approach to the efficiency of the Carnot cycle, thereby increases the generation of kWh, as indicated above, and consequently reduces the need for the often unprofitable condensing kWh; regeneration as such cannot in many cases raise the overall efficiency of the industrial thermal power plants under consideration, but it redistributes the total heat consumption at the thermal power plant, increasing the share of work at the expense of decreasing the share of heat consumers. Therefore, when it is necessary to strive to reduce the generation of condensing kWh, the use of regeneration at an industrial thermal power plant is expedient; but with large heat consumption and a limited need for electric power in the case of an isolated station, abandonment of regeneration is quite appropriate.

In the preceding exposition it was not the value of the efficiency \(\eta\) that was determined, but the reciprocal value \(\frac{1}{\eta}\), which is due not only to the fact that for \(\frac{1}{\eta}\) a simpler expression is obtained, but also to the fact that the quantity \(\frac{1}{\eta}\) enters directly into the determination of fuel consumption.

If we denote by \(b_{\mathrm{mcal}}\) the specific consumption of standard fuel for one megacalorie generated at the thermal power plant, then

\[ b_{\mathrm{mcal}} = \frac{(860 \cdot \mathcal{E} + W)\cdot 10^6} {7000(860 \cdot \mathcal{E} + W)\eta} = 143 \cdot \frac{1}{\eta}\ \mathrm{kg/mcal}. \]

In Fig. 3 are plotted the variations of the reciprocal of the coefficient of useful action \(\frac{1}{\eta}\), and hence also of \(b_{\mathrm{mcal}}\), as a function of the variation of \(\frac{W}{860 \cdot \mathcal{E}}\).

for one particular example of an installation, when \(p_1=28\ \text{ata};\ t_1=375^\circ\text{C};\ p_2=0.05\ \text{ata};\) extraction \(p_{\text{ext}}=1.5\ \text{ata};\ \eta_i=0.80;\ \eta_{\text{ku}}=0.82;\ \eta_{\text{loss}}=0.94.\) This installation is compared in Fig. 3 with a separate installation having the same initial and final parameters. Finally, for comparison, curves are plotted for back-pressure installations operating with the same initial parameters and having different back pressures: \(1.5;\ 3.0;\ 5.0\ \text{ata}\). The curves were constructed on the basis of formulas (5-a, 6, and 7). From Fig. 3 it is clear that

Fig. 3

an installation with intermediate steam extraction gives a higher efficiency for all values

\[ \frac{W}{860\cdot Э}\le 4.5. \]

For values

\[ \frac{W}{860\cdot Э}\ge 2 \]

the back-pressure installation, at \(p_{\text{ext}}=1.5\ \text{ata}\), is more advantageous than the separate installation. This shows that the field of application of back-pressure installations is not as narrow as is often thought: a back-pressure installation may fail to utilize a considerable part of the steam and nevertheless be more advantageous thermally than a separate installation. When back-pressure installations are used, measures must be taken to preserve the condensate; therefore the use of atmospheric condensers for the unutilized part of the steam is shown here. If the quantity of heat utilized for heating the heat-transfer medium per each kWh is denoted by

\[ q=\frac{W}{Э_m}, \]

then formula (5) may be rewritten in the following form:

\[ \frac{1}{\eta} = \frac{1}{\eta_m} \left[ 1+ \frac{ \left(1-\frac{Э_m}{Э}\right) \left(\frac{\eta_m}{\eta_k\ \text{or}\ \eta_p}-1\right) }{ 1+\frac{W}{860\cdot Э} } \right] = \]

\[ = \frac{1}{\eta_m} \left[ 1+ \frac{ \left(1-\frac{W}{q\cdot Э}\right) \left(\frac{\eta_m}{\eta_k\ \text{or}\ \eta_p}-1\right) }{ 1+\frac{W}{860\cdot Э} } \right] = \]

\[ = \frac{1}{\eta_m}\left\{1+\left(\frac{\eta_m}{\eta_k\ \text{or}\ \eta_p}-1\right) \left[\frac{\dfrac{860}{q}+1}{1+\dfrac{W}{860\cdot E}}-\frac{860}{q}\right]\right\}, \tag{6} \]

where \(\eta_p\) is the economic efficiency of the back-pressure engine:

\[ \eta_p=\eta_{ky}\cdot\eta_{pot}\cdot\eta_i\cdot\eta_{to}^{n}\cdot\eta_{mech}\cdot\eta_{gen}. \]

The quantity \(q\) may be determined from the formula

\[ q=\frac{860}{\eta_E\cdot h_a}\left(i_1-\eta_i\cdot h_a-t_{cond}\right)\ \frac{\mathrm{cal}}{\mathrm{kWh}}, \tag{7} \]

where \(h_a\) is the adiabatic drop up to the point of steam extraction, \(\eta_E\) is the relative electrical efficiency, and \(t_{cond}\) is the temperature of the condensate of the heating (exhaust) steam. It is precisely this formula (6) that is used to construct the curves in Fig. 3. In Fig. 3 the curve with steam extraction refers to the case in which, at

\[ \frac{W}{860\cdot E}\simeq 4.5, \]

the turbine with intermediate extraction is completely converted into a back-pressure one, so that no steam at all enters the condenser. Such a case does not occur in practice. The admission of a minimum quantity of steam to the condenser (not less than 10%) is considered absolutely mandatory in all cases. Therefore in Fig. 3 the curves corresponding precisely to this case are drawn with a dotted line. They show that, beginning with the value

\[ \frac{W}{860\cdot E}\simeq 4.1, \]

the back-pressure installation, in terms of its efficiency, already becomes just as advantageous as the installation with intermediate steam extraction, which is moreover more complex and expensive.

From formula (6) it is evident that the relation between \(\frac{1}{\eta}\) and \(\frac{W}{860\cdot E}\) is hyperbolic; moreover, for

\[ \frac{W}{860\cdot E}=\infty \]

\[ \frac{1}{\eta}=\frac{1}{\eta_m}\left\{1-\left(\frac{\eta_m}{\eta_k\ \text{or}\ \eta_p}-1\right)\frac{860}{q}\right\}, \]

at which distance from the origin there must pass the asymptote parallel to the axis of abscissas. When \(\frac{1}{\eta}=\infty\), the denominator

\[ 1+\frac{W}{860\cdot E}=0, \]

whence

\[ \frac{W}{860\cdot E}=-1, \]

i.e., at this constant distance there will be located the second asymptote, parallel to the \(\frac{1}{\eta}\) axis. The necessary points for constructing the hyperbola (6) are obtained by putting

\[ \frac{W}{860\cdot E}=0, \]

\[ \frac{1}{\eta} = \frac{1}{\eta_m} \left\{ 1+ \left( \frac{\eta_m}{\eta_k\ \text{or}\ \eta_p} \right) \left[ \frac{860}{q}+1-\frac{860}{q} \right] \right\} = \frac{1}{\eta_k\ \text{or}\ \eta_p}. \]

From what has been said it is evident that the position of the hyperbolas on the drawing will be determined, for a given extraction pressure, by the position of the asymptote parallel to the axis of abscissas, and also by the values of the efficiencies \(\eta_k\) or \(\eta_p\). Since in the case of a back-pressure installation \(\eta_p<\eta_k\), the initial points

hyperbolas on the \(1/\eta\) axis, and the hyperbolas themselves lie above the hyperbolas for plants with intermediate steam extraction, as is also seen in Fig. 3. But since \(\eta_m/\eta_n > \eta_m/\eta_k\), the asymptotes parallel to the abscissa axis lie lower for back-pressure plants than for plants with intermediate steam extraction; therefore the hyperbolas of the former will intersect the hyperbolas of the latter, as Fig. 3 also demonstrates for the case \(p_{otb}=1.5\) ata.

Such an intersection will occur when

\[ \frac{1}{\eta_m} \left[ 1+ \frac{ \left(1+\frac{E_m}{E}\right) \left(\frac{\eta_m}{\eta_n}-1\right) }{ 1+\frac{W}{860\cdot E} } \right] = \frac{1}{\eta_m^{ok}} \left[ 1+ \frac{ \frac{\eta_m^{ok}}{\eta_k}-1 }{ \frac{W}{860\cdot E}+1 } \right]. \]

Taking, for simplicity, the case when \(\frac{1}{\eta_m}\simeq \frac{1}{\eta_m^{ok}}\), we obtain, substituting

\[ E_m=\frac{W}{q}, \qquad \frac{W}{860\cdot E} = \frac{q}{860} \left[ 1- \frac{\eta_m/\eta_k-1}{\eta_m/\eta_n-1} \right]. \]

With an increase in the initial steam parameters, \(\eta_k\) and \(\eta_n\) will increase, and the increase in \(\eta_n\) will proceed faster than that of \(\eta_k\); hence the denominator will decrease faster than the numerator, and therefore the expression in square brackets will decrease faster, and consequently \(\frac{W}{860\cdot E}\) will also be smaller. With an increase in back pressure, the quantity \(q\) will increase, which will affect the increase of \(\frac{W}{860\cdot E}\).

Let us note that the hyperbolas of Fig. 3 give us a picture of the variable operating regime of the plants, a picture of the change in the reciprocal value \(1/\eta\) of the efficiency, and therefore also of the fuel consumption, as a function of the change in the energy coefficient \(\frac{W}{860\cdot E}\), which is a characteristic indicator of the power supply of the territory under consideration.

Reciprocal Value for the Efficiency of a Binary Plant

If from formula (5) the efficiencies of plants with an independent boiler house, with back pressure, and purely condensing plants are derived as special cases, then formula (5) itself is a special case derived from a more general efficiency formula for binary plants. Denoting the total heat consumption for a binary plant by \(Q_b\), the heat consumption by the mercury part by \(Q_{pm}\), and, as above, the same for the steam turbine by \(Q_k\) and \(Q_m\), we obtain

\[ Q_b=Q_{pm}+Q_k+Q_m. \]

Let \(E_{pm}\) be the production of kWh by the mercury part of the plant over the time interval under consideration, \(\eta_{pm}\) the efficiency of the mercury part, and \(\eta_b\) the same for the entire binary plant as a whole. Then

\[ \frac{860\cdot E+W}{\eta_b} = \frac{860\cdot E_{pm}}{\eta_{pm}} + \frac{860\left[E-\left(E_{pm}+E_m\right)\right]}{\eta_k} + \frac{860\cdot E_m+W}{\eta_m} \tag{8} \]

In this case

\[ \eta_{\rho m}=\eta_{ky}^{\rho T}\cdot \eta_{nom}^{\rho T}\,\eta_{mex}^{\prime\prime}\cdot \eta_{gen}^{\prime\prime}; \tag{9} \]

\[ \eta_{mex}^{\prime\prime}\cdot \eta_{gen}^{\prime\prime} = 1-\left(1-\eta_{mex}^{\rho T}\cdot \eta_{gen}^{\rho T}\right) \cdot \frac{\eta_i^{\rho T}\cdot h_a^{\rho T}} {i^{\rho T}-i_{cond}^{\rho T}}; \tag{9-a} \]

\[ \eta_m= \eta_{ky}^{\rho T} \left[ \varepsilon+(1-\varepsilon)\eta_{nom}^{\rho m}\cdot \eta_{mex}^{\prime\prime}\cdot \eta_{gen}^{\prime\prime} \right] \cdot \eta_{nom}^{vod}\cdot \eta_{mex}^{\prime}\cdot \eta_{gen}^{\prime}; \tag{9-b} \]

\[ \eta_k= \eta_{ky}^{\rho T} \left[ \varepsilon+(1-\varepsilon)\eta_{nom}^{\rho T}\cdot \eta_{mex}^{\prime\prime}\cdot \eta_{gen}^{\prime\prime} \right] \cdot \eta_{nom}^{vod}\cdot \eta_i^{vod}\cdot \eta_i^{vod\prime}\cdot \eta_{mex}^{\prime}\cdot \eta_{gen}^{\prime}, \tag{9-c} \]

where \(\eta_{nom}^{\rho T}\) is the efficiency of transport of the mercury heat flow from the boiler through the steam pipe, the mercury turbine, and the feed system, and \(\eta_{nom}^{vod}\) is the same for the steam-water part and the evaporator-condenser, through the steam pipe, the steam turbine, its condenser, and the feed system back to the evaporator-condenser; \(\varepsilon\) is the fraction of steam obtained from the boiler, besides the evaporator-condenser, from the water screens. From formula (3) we have:

\[ \frac{1}{\eta_b} = \frac{1}{\eta_m} + \frac{1}{1+\dfrac{W}{860\cdot \mathcal{E}}} \left\{ \left(\frac{1}{\eta_k}-\frac{1}{\eta_m}\right) \left(1-\frac{\mathcal{E}_m}{\mathcal{E}}\right) - \frac{\mathcal{E}_{\rho m}}{\mathcal{E}} \left(\frac{1}{\eta_k}-\frac{1}{\eta_{\rho T}}\right) \right\}. \tag{10} \]

In Fig. 4 a nomogram is given for determining \(\eta_b\). In the absence of the mercury part, \(\mathcal{E}_{\rho T}=0\), and then expression (10) is transformed into formula (5). Thus we can consider the binary plant as a heat-power plant with a back-pressure mercury part and with a steam turbine—

Fig. 4

bine, having regeneration and intermediate steam extraction for external heat consumption. From formula (9) it is clear that: 1) an increase in \(\dfrac{W}{860\cdot E}\) increases \(\eta_b\); 2) an increase in \(\dfrac{E_{pT}}{E}\) raises \(\eta_b\); 3) an increase in \(E_m/E\) also raises \(\eta_b\).

From formula (9) it is clear that deepening the vacuum in the condenser-evaporator, on the one hand, by increasing \(E_{pT}\), also increases \(\eta_b\), but on the other hand, by decreasing \(E_m\), lowers \(\eta_b\). Such a contradictory effect of changing the vacuum in the condenser-evaporator leads to the necessity of constructing, on the basis of formula (9), a curve of the dependence of \(\eta_b\) on the pressure \(P_2^T\) in the condenser-evaporator, in order to be able to determine, from the maximum \(\eta_b\), the corresponding \(P_2^T\).

Formula (5) can be successfully used to determine the efficiency of an entire system that includes a number of district-heating and condensing units; for this it is necessary to know only the total generation of electric power \(E\), kWh, over a definite interval of time, the value \(W\) over the same period, the weighted-average efficiency values \(\eta_k\) for the whole system, and also the generation \(E_m\) for heat consumption by the whole system.

Reciprocal of the Efficiency for a Power System with Hydroelectric Stations and Independent Boiler Houses

With the development in the USSR of large power systems that include not only thermal electric power stations, but also hydroelectric stations, as well as independent separate boiler houses, it is very important to have a more generalized expression for the efficiency of such a complex system and to establish the interrelation among the influencing factors. If \(\eta\) is the efficiency of the thermal part of the power system, \(E_{hidr}\) is the supply to the system, in kWh, from hydroelectric stations, and \(W_k\) is the number of calories produced over the period of time under consideration (for example, a year) for heat consumers in independent heating boiler houses, then, taking \(\eta_{ku}\) as the weighted-average efficiency of the independent boiler houses, one may write:

\[ \frac{860(E-E_{hidr})+W+W_k}{\eta} = \frac{860\cdot E_m+W}{\eta_m} + \frac{860(E-E_{hidr}-E_m)}{\eta_k} + \frac{W_k}{\eta'_{ku}\cdot\eta'_{pot}} \tag{11} \]

From equation (11) we obtain:

\[ \frac{1}{\eta} = \frac{1}{1+\left(1+\frac{W_k}{W}\right)\frac{W}{860(E-E_{hidr})}} \cdot \frac{1}{\eta_m} \left\{ \frac{\eta_m}{\eta_k} - \frac{E_m}{E-E_{hidr}} \left(\frac{\eta_m}{\eta_k}-1\right) + \left[ \frac{W_k}{W}\cdot \frac{\eta_m}{\eta'_{ku}\cdot\eta'_{pot}} +1 \right] \frac{W}{860(E-E_{hidr})} \right\} \tag{12} \]

From the last formula it follows that:

1) if \(E_{hidr}=0\), then formula (12) becomes

\[ \frac{1}{\eta} = \frac{1}{1+\left(1+\frac{W_k}{W}\right)\frac{W}{860E}} \cdot \frac{1}{\eta_m} \left\{ \frac{\eta_m}{\eta_k} - \frac{E_m}{E} \left(\frac{\eta_m}{\eta_k}-1\right) + \left[ \frac{W_k}{W}\cdot \frac{\eta_m}{\eta'_{ku}\cdot\eta'_{pot}} +1 \right] \cdot \frac{W}{860\cdot E} \right\} \]

The obtained formula (13) corresponds to a system without hydroelectric stations, but with independent boiler houses.

2) If \(\mathcal{E}_{\text{hydr}}=0\) and \(W_k=0\) (Fig. 5), then we obtain the formula (5) already presented above.

3) An increase in \(W\)—the number of calories used by heat consumers from extraction steam of the engines, with a simultaneous increase in \(\mathcal{E}_t\), increases the efficiency \(\eta\) of the system.

4) Any increase in \(W_k\)—the heat supplied by independent heating boiler houses—while the total heat consumption

Fig. 5

\(W+W_k=\mathrm{const}\) decreases \(\eta\)—the efficiency of the system, since in this case \(\mathcal{E}_t\)—the generation from heat consumption—will fall. Despite what has just been said, independent boiler houses for cutting off peaks of the heat load are now being designed for inclusion in our power systems, in order to increase the utilization factor of the equipment at CHP plants.

5) The expression inside the braces of (12) can be represented, knowing that \(W=q\cdot \mathcal{E}_t\), in the following form:

\[ \left\{ \frac{\eta_t}{\eta_k} + \left[ \left( \frac{W_k}{W}\cdot \frac{\eta_t}{\eta'_{ku}\cdot \eta'_{pot}} +1 \right) \frac{q}{860} - \left( \frac{\eta_t}{\eta_k}-1 \right) \right] \frac{\mathcal{E}_t}{\mathcal{E}-\mathcal{E}_{\text{hydr}}} \right\} = \]

\[ = \left\{ \frac{\eta_t}{\eta_k} + \varphi\cdot \frac{\mathcal{E}_t}{\mathcal{E}-\mathcal{E}_{\text{hydr}}} \right\}, \quad \text{where } \varphi= \left[ \left( \frac{W_k}{W}\cdot \frac{\eta_t}{\eta'_{ku}\cdot \eta'_{pot}} +1 \right) \frac{q}{860} - \left( \frac{\eta_t}{\eta_k}-1 \right) \right]. \]

If we now carry out the actual division of the fraction \(\dfrac{E_m}{E} \cdot \dfrac{E_m}{E_{\mathrm{hydr}}}\) by the denominator of the common factor before the braces,

\[ 1+\left(1+\frac{W_k}{W}\right)\frac{q\cdot E_m}{860(E-E_{\mathrm{hydr}})}, \]

we obtain

\[ \frac{1}{\left(1+\frac{W_k}{W}\right)\frac{q}{860}} - \frac{\dfrac{1}{(1+W_k/W)\cdot q/860}} {1+\left(1+W_k/W\right)\dfrac{q\cdot E_m}{860(E-E_{\mathrm{hydr}})}} , \]

so that

\[ \frac{1}{\eta} = \frac{1}{1+\left(1+\frac{W_k}{W}\right)\dfrac{W}{860(E-E_{\mathrm{hydr}})}} \cdot \frac{1}{\eta_m} \left\{ \frac{\eta_m}{\eta_k} - \frac{860\varphi}{\left(1+\dfrac{W_k}{W}\right)q} \right\} + \frac{860\varphi}{\eta_m\left(1+\dfrac{W_k}{W}\right)q}. \tag{14} \]

It follows clearly from this that any increase in \(E_{\mathrm{hydr}}\) increases the efficiency \(\eta\) of the system, which occurs through the displacement of condensing kWh. By increasing \(E_{\mathrm{hydr}}\), we can offset the reduction in efficiency \(\eta\) caused by the inclusion of independent boiler houses, since \(E_{\mathrm{hydr}}\) will displace condensing kWh.

Formula (14) establishes that the change of \(1/\eta\) as a function of

\[ \frac{W+W_k}{860(E-E_{\mathrm{hydr}})} \]

takes place according to the law of a hyperbola, whose asymptotes are, on the one hand, a straight line parallel to the axis of abscissas and lying from it at a distance

\[ \frac{1}{\eta_m}-\frac{860\varphi}{(W_k/W)q}, \]

and, on the other hand, a straight line parallel to the axis \(1/\eta\) and lying from it at a distance \(-1\), for when \(1/\eta=\infty\) it must be

\[ 1+\frac{W+W_k}{860(E-E_{\mathrm{hydr}})}=0. \]

Let us note that an increase in \(E_{\mathrm{hydr}}\) increases

\[ \frac{W+W_k}{860(E-E_{\mathrm{hydr}})}, \]

thereby decreasing \(1/\eta\), i.e., the specific fuel consumption.

In accordance with § 1059 of the Rules for the Technical Operation of Stations and Networks, the electrical load of district-heating turbines corresponding to their heat load must be assigned to the base part of the system’s total load curve. For hydroelectric stations (§ 1062) having a basin for daily regulation of runoff, during periods of low water discharge, which in our country occurs mainly in the winter period, as a rule the corresponding peak part of the total load curve should be assigned; during periods of high water discharge (§ 1061), which usually coincide with the spring–summer period of the year, hydroelectric stations should be assigned the base part of the total daily load curve of the system. Ta-

Thus, it turns out that both “forced” schedules (for heat consumption and hydroelectric generation) in most cases successfully complement one another. A separate calculation of the cost of a kWh of hydroelectric stations when they operate on base load, as is often done, distorts the actual magnitude of the cost of a kWh. Since the hydroelectric station must operate on base load, then, consequently, the thermal stations, which are organically necessary in the system, are forced to operate in the semi-peak and peak-load region, which, of course, worsens their operating conditions. Therefore the separate calculation of the cost of an isolated kWh of hydroelectric stations is incorrect: it must be calculated together with the cost of the thermal kWh organically connected with it, considering the system as a whole and being guided by the indicated §§ 1059--1061 of the “Rules for Thermal Power Plants.” Among the components of the cost of a kWh of the system, a prominent place belongs to fuel consumption, determined from the efficiency of the system with the application of our formula (12)¹.

Reciprocal of the efficiency under variable operating conditions

The efficiency of a turbine installation with combined production of heat and electric power will have the following expression in the case of variable operation with one steam extraction point:

\[ \eta_{mx}=\frac{860\cdot N_x+W_x}{Q_x} =\frac{860\cdot N_x+W_x}{i'_1\cdot D_x}. \]

Here

\[ W_x=\eta_{\text{boil}}(i_{\text{ext}}-t_{\text{cond}})D_{\text{ext}}, \]

\[ D_x=a_1+b_1N_x+\left(1-\frac{\eta_1h_1}{\eta_{\mathrm{e}}H}\right)D_{\text{ext}}, \]

\[ i'_1=i_1-t_{\text{cond}}. \]

Thus

\[ \eta_m= \frac{860N_x+(i_{\text{ext}}-t_{\text{cond}})\eta_{\text{boil}}\cdot D_{\text{ext}}} {i'_1\left[a_1+b_1N_x+\left(1-\frac{\eta_1h_1}{\eta_{\mathrm{e}}H}\right)D_{\text{ext}}\right]}. \]

In determining the fuel consumption one must use the reciprocal value of the efficiency, which, therefore, will have the form:

\[ \frac{1}{\eta_{mx}}= \frac{ i'_1\left[ a_1+b_1N_x+\left(1-\frac{\eta_1\cdot h_1}{\eta_{\mathrm{e}}\cdot H}\right)D_{\text{ext}} \right] } {860\cdot N_x+(i_{\text{ext}}-t_{\text{cond}})\eta_{\text{boil}}\cdot D_{\text{ext}}} = \]

\[ = \frac{A_1+B_1N_x+C_1D_{\text{ext}}} {860\cdot N_x+C_2D_{\text{ext}}} = \frac{B_1}{860} + \frac{ A_1-\left(\frac{B_1C_2}{860}-C_1\right)D_{\text{ext}} } {860\cdot N_x+C_2D_{\text{ext}}} \]

If \(D_{\text{extr}}=0\), we have a condensing installation, and

\[ \frac{1}{\eta_{m x}^{k}}=\frac{B_{1}}{860}+\frac{A_{1}}{860\cdot N_{x}}, \]

that is, \(1/\eta_{m x}^{k}\) varies as a function of \(N_x\) according to the law of a hyperbola, whose asymptotes are the axis \(1/\eta_{m x}\) and a straight line parallel to the axis of abscissas \((N_x)\), drawn at a distance \(B_1/860\) from the origin.

On the other hand, if the installation operates as a back-pressure installation with complete utilization of the exhaust heat, we have:

\[ \frac{1}{\eta_{m x}^{p.d.}} = \frac{D_{x}^{p.d.}\cdot i_{1}^{\prime}} {860\cdot N_x+(i_{\text{extr}}-t_{\text{cond}})\eta_{\text{boil}}\cdot D_{x}^{p.d.}} = \]

\[ = \frac{i_{1}^{\prime}(a_{1}^{\prime}+b_{1}^{\prime}N_x)} {860\cdot N_x+(i_{\text{extr}}-t_{\text{cond}})\cdot\eta_{\text{boil}}\cdot(a_{1}^{\prime}+b_{1}^{\prime}N_x)} . \]

The quantity of steam entering the condenser in an installation with intermediate extraction\(^1\) is

\[ D_k=a_1+b_1N_x-\frac{\eta_1 h_1}{\eta_E H}\cdot D_{\text{extr}} . \]

In the case of the back-pressure regime \(D_k=0\), and consequently,

\[ D_{\text{extr}}=D_x^{p.d.} = \left(a_1+b_1N_x\right)\frac{\eta_E\cdot H}{\eta_1\cdot h_1} = a_1^{\prime}+b_1^{\prime}\cdot N_x . \]

Therefore

\[ \frac{1}{\eta_{m x}^{p.d.}} = \frac{i_{1}^{\prime}(a_{1}^{\prime}+b_{1}^{\prime}N_x)} {(i_{\text{extr}}-t_{\text{cond}})\cdot\eta_{\text{boil}}\cdot a_{1}^{\prime} + \left[860+b_{1}^{\prime}\cdot\eta_{\text{boil}}\cdot(i_{\text{extr}}-t_{\text{cond}})\right]N_x} = \]

\[ = \frac{i_{1}^{\prime}\cdot b_{1}^{\prime}} {860+b_{1}^{\prime}\cdot\eta_{\text{boil}}(i_{\text{extr}}-t_{\text{cond}})} + \]

\[ + \frac{860\cdot i_{1}^{\prime}\cdot a_{1}^{\prime}} {\left[860+b_{1}^{\prime}\cdot\eta_{\text{boil}}(i_{\text{extr}}-t_{\text{cond}})\right]} \times \]

\[ \times \frac{1} {(i_{\text{extr}}-t_{\text{cond}})\cdot\eta_{\text{boil}}\cdot a_{1}^{\prime} + \left[860+b_{1}^{\prime}\cdot\eta_{\text{boil}}\cdot(i_{\text{extr}}-t_{\text{cond}})\right]N_x}. \tag{16} \]

Thus, in the case of a back-pressure installation with final utilization of heat, \(1/\eta_{m x}^{p.d.}\) also varies as a function of \(N_x\) according to the law of a hyperbola, whose asymptotes are, on the one hand, a straight line parallel to the axis \(N_x\), drawn from the origin at a distance

\[ \frac{i_1^{\prime}b_1^{\prime}} {860+b_1^{\prime}\cdot\eta_{\text{boil}}\cdot(i_{\text{extr}}-t_{\text{cond}})}, \]

\(^1\) This expression is derived from the condition that a turbogenerator operating with intermediate steam extraction can provide the same output when it also operates in the condensing regime, i.e. when

\[ (a_1+b_1N_x)\eta_E\cdot H = D_{\text{extr}}\cdot\eta_1 h_1 + D_k\cdot\eta_E\cdot H, \]

whence

\[ D_k=a_1+b_1N_x-\frac{\eta_1h_1}{\eta_E\cdot H}D_{\text{extr}} . \]

On the other hand, the straight line at a distance

\[ N'_x=-\,\frac{a'_1\left(i_{\mathrm{ext}}-t_{\mathrm{cond}}\right)\eta_{\mathrm{boil}}} {860+b'_1\eta_{\mathrm{boil}}\left(i_{\mathrm{ext}}-t_{\mathrm{cond}}\right)} . \]

from the origin of the powers \(N_x\).

These two hyperbolas—the condensing-plant and back-pressure-plant hyperbolas—limit the extreme bounds between which changes of

\[ \frac{1}{\eta_{mx}} \]

with respect to \(N_x\) will occur already under the operating regime of a plant with intermediate steam extraction, for varying values of \(D_{\mathrm{ext}}\) and \(N_x\). Turning to formula (15), one can see that, for a certain value of \(D_{\mathrm{ext}}\), when

\[ A_1=\left(\frac{B_1 C_2}{860}-C_1\right)D_{\mathrm{ext}}, \]

that is, when

\[ D_{\mathrm{ext}}=\frac{A_1}{B_1C_2/860-C_1}, \quad \text{then} \quad \frac{1}{\eta_{mx}}=\frac{B_1}{860}, \tag{17} \]

i.e., it proves to be a constant quantity, independent of changes in \(N_x\), so that this will be a straight line parallel to the \(N_x\) axis, drawn at a distance \(B_1/860\) from it.

In the remaining cases, however, for any given value of \(D_{\mathrm{ext}}\), changes of

\[ \frac{1}{\eta_{mx}} \]

will occur, as \(N_x\) changes, again according to the law of a hyperbola. It must be noted that for

\[ A_1>\left(\frac{B_1 C_2}{860}-C_1\right)D_{\mathrm{ext}} \]

an increase in \(N_x\) will produce a decrease in

\[ \frac{1}{\eta_{mx}}, \]

whereas for

\[ A_1<\left(\frac{B_1 C_2}{860}-C_1\right)D_{\mathrm{ext}}, \]

on the contrary, this quantity will increase; moreover, the above-mentioned straight line (17), parallel to the \(N_x\) axis, is here a line of division, as it were. All these hyperbolas will have, as one asymptote, the straight line parallel to the \(N_x\) axis and drawn at a distance \(B_1/860\) from it. The second asymptotes here are straight lines located from the origin of the \(N_x\) coordinates at distances

\[ \frac{C_2D_{\mathrm{ext}}}{860}. \]

Since the specific consumption of equivalent fuel for the generated megacalorie, in kg, is

\[ b_{mx}=\frac{10^6}{7000\eta}=143\,\frac{1}{\eta}, \]

the curves representing changes in

\[ \frac{1}{\eta} \]

as a function of \(N_x\) will at the same time also be the curves for \(b_{mx}\), as a function of \(N_x\), on a scale changed by a factor of 143.

In the appended graph (Fig. 6), the above method has been used for one particular example of a turbogenerator with the following characteristics: \(p_1=29\) ata; \(t_1=400^\circ\mathrm{C}\); \(p_2=0.04\) ata; \(p_{\mathrm{ext}}=6\) ata; tests have established that \(D_x=5+4.16N_x+0.677D_{\mathrm{ext}}\); \(N_e=5\) MW; \(\eta_{\mathrm{boil}}=0.98\).¹

¹ Figs. 6 and 7 are constructed on the assumption that \(i'_1=i_1\), i.e., that the condensate of the heating steam is not returned to the heat-and-power plant \((t_{\mathrm{cond}}=0)\).

In the case of two points of steam extraction in the turbine, the reciprocal quantity for the efficiency is

\[ \frac{1}{\eta_{mx}} = \frac{ t'_1\left[ a_1+b_1N_x+\left(1-\frac{\eta_1h_1}{\eta_{\mathrm{e}}H}\right)D'_{\mathrm{ext}} +\left(1-\frac{\eta_2h_2}{\eta_{\mathrm{e}}H}\right)D''_{\mathrm{ext}} \right] }{ 860N_x+(i'_{\mathrm{ext}}-t'_{\mathrm{cond}})\eta_{\mathrm{boil}}D'_{\mathrm{ext}} +(i''_{\mathrm{ext}}-t''_{\mathrm{ext}})\eta_{\mathrm{boil}}D''_{\mathrm{ext}} } = \]

\[ = \frac{ a_1+b_1N_x+\left(1-\frac{\eta_1h_1}{\eta_xH}\right)D'_{\mathrm{ext}} +\left(1-\frac{\eta_2h_2}{\eta_{\mathrm{e}}H}\right)D'_{\mathrm{ext}} }{ \frac{860N_x}{t'_1} + \frac{(i'_{\mathrm{ext}}-t'_{\mathrm{cond}})\eta_{\mathrm{boil}}D'_{\mathrm{ext}}}{t'_1} + \frac{(i''_{\mathrm{ext}}-t''_{\mathrm{cond}})\eta_{\mathrm{boil}}D''_{\mathrm{ext}}}{t'_1} } \tag{18} \]

Change in efficiency under variable operating conditions of a steam engine with intermediate steam extraction.

Fig. 6

Let us denote

\[ y_1=a_1+b_1N_x+\left(1-\frac{\eta_1h_1}{\eta_{\mathrm{e}}H}\right)D'_{\mathrm{ext}}, \]

i.e., \(y_1=f_1(N_x)\) for different values of \(D'_{\mathrm{ext}}\) represents a family of parallel straight lines. On the other hand,

\[ y_2=y_1+\left(1-\frac{\eta_2h_2}{\eta_{\mathrm{e}}H}\right)D''_{\mathrm{ext}}, \]

i.e., \(y_2=f_2(y_1)\) for different values of \(D''_{\mathrm{ext}}\) also represents a family of straight lines inclined to the \(y_1\) axis at an angle of \(45^\circ\).

Within the limits of the extraction pressures usually employed \((1.5—1.30\ \text{ata})\)

\[ (i'_{\text{ext}}-t'_{\text{cond}})\cong(i''_{\text{ext}}-t''_{\text{cond}}). \]

Then the denominator of expression (18) will be

\[ \eta_{\text{boil}}(i'_{\text{ext}}-t'_{\text{cond}}) \cong \eta_{\text{boil}}(i''_{\text{ext}}-t''_{\text{cond}}) \cong 510\ \text{kcal}/\text{kg}, \]

i.e., again for different values of \(N_x\) and \((D'_{\text{ext}}+D''_{\text{ext}})\) the denominator of (18) will be

\[ y_3=\frac{860N_x}{i_1'}+\frac{510}{i_1'}(D'_{\text{ext}}+D''_{\text{ext}}), \]

and we shall obtain a bundle of straight lines inclined to the \(N_x\) axis at an angle whose tangent is \(860/i_1'\). From equation (18) it follows that

\[ \frac{1}{\eta_{mx}}=\frac{y_2}{y_3}, \]

so that

\[ y_2=\frac{1}{\eta_{mx}}\cdot y_3. \]

Assigning different values to \(1/\eta_{mx}\), we obtain a bundle of straight lines in the coordinates \(y_2\) and \(y_3\), inclined to the \(y_3\) axis at different angles, whose tangent is \(1/\eta_{mx}\).

Fig. 7 gives the constructed nomogram, where in quadrant I there is a bundle of straight lines \(y_1=f_1(N_x)\); in quadrant II, a bundle of straight lines \(y_2=f_2(y_1)\);

Fig. 7

in quadrant IV—a bundle of straight lines \(y_3=f_3(N_x)\), and in quadrant III—a bundle of straight lines \(y_2=f_4(y_3)\) for different values of \(1/\eta_{mx}\).

The construction has been made for the particular example:

\[ p_1=100\ \text{ata},\quad t_1=500^\circ\text{C};\quad p_2=0.04\ \text{ata},\quad D'_{\text{ext max}}=60\ \text{t/h},\quad D''_{\text{ext max}}=110\ \text{t/h}; \quad \eta_{\text{boil}}=0.98;\quad p'_{\text{ext}}=1.2\ \text{ata},\quad p''_{\text{ext}}=13\ \text{ata},\quad N_{\text{e}}=20\ \text{MW}; \]

\[ \eta_1/\eta_3\cong\eta_2/\eta_3\cong1.0;\quad a_1=6.0;\quad b_1=3.75. \]

Other applications of the above formulas

Our formula (5), intended mainly for determining \(\eta\), the efficiency of a CHP plant or of an entire power system, may have various other applications. We shall give several examples of such applications. Thus, formula (5) gives us the possibility of approaching the solution of the question of the nature of the drive for auxiliary equipment: should it be stea-

steam or electric? This question has repeatedly been brought up for discussion in the literature, and divergent opinions have been expressed on this subject. Prof. A. A. Lagovskii, in his book [2], intended “to serve as a guide not only in the study of the corresponding course, but also in design work by students,” expresses the following considerations: “The choice of drive for the feed pump—by an electric motor or by a steam turbine—is determined by the possibility of utilizing the exhaust steam of the latter. If such utilization is limited, then an electric pump is chosen as the principal working pump. If, however, the exhaust steam is used continuously in the station’s heat economy, then it is quite rational to use turbo-pumps.” Meanwhile, if one proceeds from our formula (5), this question looks different. Indeed, in the case of a purely condensing station (without regeneration), it is expedient to convert it partially into a combined heat-and-power plant, creating a heat consumption at least for heating the feed water by the exhaust steam of the turbines of the auxiliary mechanisms, obtaining at least a small output for heat consumption \((E_t)\), in this case in the form of mechanical energy of the steam drive. In the same condensing station, but with regenerative heating of the feed water by extraction steam from intermediate stages of the main turbines, the use of exhaust steam in the turbines of auxiliary mechanisms will already be unprofitable, since the internal relative efficiency of small steam turbines usually does not exceed 60% at economical operation (often it is lower), whereas for large main turbines it is 80–84%, i.e., exceeds the former by 1.3–1.4 times, which more than covers the losses in the network and in the motor of the electric drive, not to mention that regeneration increases the efficiency of the cycle of the theoretical machine. Thus, with the use of the exhaust of auxiliary turbines the efficiency would be reduced. In the case of a combined heat-and-power plant with the use of heat not only for the station’s internal consumption, but also for outside consumers, utilization of the exhaust of auxiliary turbines can be profitable only under such circumstances when there is insufficient heat from the main turbines at condensing output in kWh reduced to the permissible minimum. The installation of a turbine for station service, of greater capacity than the individual turbines at each auxiliary mechanism, ensures obtaining a higher efficiency \(\eta\) of the CHP plant owing to the higher internal efficiency of such a turbine, making it possible to obtain a larger output \((E_t)\) for heat consumption. But it is clear that this internal efficiency of the station-service turbine is nevertheless lower than that of the main turbines with regeneration, which is why installation of the former is usually not advantageous in a thermal respect.

As a second example of the possibility of using formula (5), let us point to the choice of the type of heaters and the arrangement of the condensate lines of the heating steam of various heaters. This question is treated in the book Standardization of Steam Turbines and the Thermal Schemes of Large Power Stations [3], recommended in the curricula as a textbook for power-engineering higher schools. Explaining the disadvantages of surface heaters, the author writes: “The quantity \(\Delta i\) is the specific thermal loss of the surface method of transferring heat from the heating steam to the main condensate stream, since in this process, neglecting losses to the surroundings, although all the heat passes quantitatively into this stream, qualitatively, by lowering its temperature, it becomes less valuable (with the same heating steam, but with mixing heaters the feed water would enter the boiler hotter by \(\Delta i\)).” And further: “As is evident, scheme D is the least economical. This results from the fact that in scheme D the hot condensate of the heating steam of all the heaters, instead of heating the main condensate stream in the region …”

high temperature is used for its heating in the region of low temperature. Therefore what is saved is not high-value steam from the earlier extractions, but the least valuable steam from the coldest extraction. From the standpoint of the second law of thermodynamics, the matter appears quite differently from what is stated in the quoted passages from the aforementioned book; it amounts to the fact that, in the presence of surface heaters, the entropy of the steam exhausted to the condenser increases and the heat loss to the cold source grows.

One or another direction of heater drainage entails different quantities of steam extraction for each of them; moreover, in the case, for example, of collecting all the condensate of relatively high temperature by means of a main pipeline in the low-pressure heater, the steam extraction for the latter will be considerably smaller, whereas in the higher-pressure heaters more steam will be extracted because condensate from the heaters above them does not enter them, as occurs with the cascade method of drainage. As a result, the area of the theoretical regenerative cycle will be somewhat reduced because of the displacement of the intermediate adiabats of the stepped expansion line into the interior of the cycle, beginning with the second heater, if counted from the side of the high-pressure heater. But, on the other hand, the area of the cycle will increase as a result of the displacement of the last adiabat of steam expansion after the last low-pressure heater in the direction opposite to the displacement of the above-mentioned adiabats, so that the total area of the cycle will not change; however, the entropy after the final expansion will increase and, consequently, the heat rejection to the cold source will increase, and the losses will grow.

When steam transformers are used, we have a reduction in the generation $\mathcal{E}_m$ kWh for heat consumption, and hence also in the efficiency of the heat-and-power plant, in comparison with the case of direct extraction of steam for production; this is precisely due to the reduction of the heat drop in the turbine up to the extraction point, a reduction caused by the specific loss $\Delta i$ in the steam transformers.

Everything stated in the last paragraph gives grounds for asserting that our formulas (5) and (12) make it possible to carry out a comparative evaluation of one or another station heat scheme according to its effect on the efficiency of the heat-and-power plant and, consequently, on fuel consumption, bearing in mind the inclusion in the scheme not only of heaters and steam transformers, but also of evaporators, deaerators, etc.

We can also use our formula (5) to determine the efficiency $\eta_{par}$ of the steam pipeline of a heat-and-power plant. If the heat-and-power plant operated without a steam pipeline at the initial steam parameters corresponding to the boiler-house parameters, then we would have the efficiency of the heat-and-power plant $\eta_1'$. In the case where a steam pipeline is present, this efficiency will be $\eta_1'' < \eta_1'$, so that the efficiency of the steam pipeline is

\[ \eta_{par}=\eta_1''/\eta_1' = \frac{ 1+\dfrac{(1-\mathcal{E}_m'/\mathcal{E})(\eta_m/\eta_k' - 1)} {\dfrac{W}{860\mathcal{E}}+1} }{ 1+\dfrac{(1-\mathcal{E}_m''/\mathcal{E})(\eta_m/\eta_k'' - 1)} {\dfrac{W}{860\mathcal{E}}+1} } = \frac{ \dfrac{W}{860\mathcal{E}}+1+ \left(1-\dfrac{\mathcal{E}_m'}{\mathcal{E}}\right) \left(\dfrac{\eta_m}{\eta_k'}-1\right) }{ \dfrac{W}{860\mathcal{E}}+1+ \left(1-\dfrac{\mathcal{E}_m''}{\mathcal{E}}\right) \left(\dfrac{\eta_m}{\eta_k''}-1\right) }, \tag{19} \]

where $\mathcal{E}_m'$ is the generation for heat consumption in the absence of a steam pipeline,

\(E_m'\)—the same when it is present.

It follows from this that the efficiency \(\eta_{\mathrm{par}}\) depends on the variable quantity of production

\[ M=\left(1-\frac{E_m'}{E}\right)\left(\frac{\eta_m}{\eta_k''}-1\right), \tag{20} \]

since all the other quantities are either specified by the conditions of the district energy supply \((E\) and \(W/E)\), or by the steam parameters in the boiler \((\eta_k')\), or else by these latter and by the parameters of the extraction steam \((E_m')\). In his articles devoted to the question of determining the diameter of the passage of steam lines according to their maximum efficiency, the author established that \(M=f(d)\), a function of the internal diameter of the steam-line passage, so that only \(d\) can be selected in such a way that the value of the efficiency \(\eta_{\mathrm{par}}\) of the steam line is obtained close to its maximum, bearing in mind that at the maximum of \(\eta_{\mathrm{par}}\) the diameters \(d\) turn out to be excessively large and the steam velocities understated, which, of course, increases the cost of the steam line.

The efficiency of the steam line \(\eta_{\mathrm{par}}\) will be a maximum when the denominator is a minimum, i.e., when the product

\[ M=\left(1-\frac{E_m''}{E}\right)\left(\frac{\eta_m}{\eta_k''}-1\right) \]

turns out to be minimal. But

\[ \frac{E_m''}{E}=\frac{W}{E}\cdot\frac{1}{q}=R\cdot\frac{1}{q}, \]

if it is assumed that \(W=E_m''\cdot q\), where \(q\) is the number of calories obtained from each kWh in heat consumption, and \(R=\frac{W}{E}\) is, as indicated, a specified quantity. As can be seen from Fig. 8,

\[ q=\frac{W}{E_m''}=d_z''\left(i_n''-t_{\mathrm{cond}}\right) =\frac{860\left(i_n'-t_{\mathrm{cond}}\right)}{\eta_z''\left(i_1'-i_n'\right)} = \]

\[ =\frac{860\left(i_n'+\Delta i_n'-t_{\mathrm{cond}}\right)} {\eta_z''\left(i'-\Delta i'-i_n'-\Delta i_n'\right)}. \]

Further, \(\Delta i_n'=x\tan\beta\) and \(y=x\tan\beta_1\), so that \(\Delta i_n'=y\tan\beta/\tan\beta_1\). In addition,

\[ \frac{y}{\Delta'' i'}= \frac{\Delta'' i'\cdot\tan\beta_1-\Delta' i'\cdot\tan\beta_2} {\Delta'' i'\cdot\tan\beta_1}. \]

Approximately \(\beta_1\simeq\beta_2\), whence \(y=\Delta'' i'-\Delta' i'\), and consequently,

\[ \Delta i_n'=(\Delta'' i'-\Delta' i')\tan\beta/\tan\beta_1 =\alpha(\Delta'' i'-\Delta' i'). \]

Therefore

\[ \frac{1}{q}=\frac{E_m''}{W} = \frac{\eta_z''\left[i'-i_n'-\Delta' i'-\alpha(\Delta'' i'-\Delta' i')\right]} {860\left[i_n'+\alpha(\Delta'' i'-\Delta' i')-t_{\mathrm{cond}}\right]}; \]

\[ 1-\frac{E_m''}{E} = 1-\frac{W}{E}\cdot\frac{1}{q} = \]

\[ =1-\frac{W}{E}\cdot\frac{\eta_z''}{860}\cdot \frac{i'-i_n'-\Delta' i'-\alpha(\Delta'' i'-\Delta' i')} {i_n'+\alpha(\Delta'' i'-\Delta' i')-t_{\mathrm{cond}}}. \]

Denoting

\[ \frac{W}{\mathcal{E}}\cdot \frac{\eta_{\mathrm{e}}^{\prime\prime}}{860}=\mu \]

and carrying out division of the numerator polynomial by the binomial denominator at the subtrahend, we obtain

\[ 1-\frac{\mathcal{E}_m}{\mathcal{E}} = 1-\mu\left[ \frac{i'-\Delta'i'-t_{\mathrm{cond}}} {i'_n+\alpha(\Delta''i'-\Delta'i')-t_{\mathrm{cond}}} +1 \right]. \tag{21} \]

If we now turn to the purely condensing part of the installation, then, in the presence of a steam line, we shall have (Fig. 9) its efficiency

\[ \eta_k^{\prime\prime} = \frac{(i'_1-i_k^{\prime\prime})\eta_{\mathrm{e}}^{\prime\prime}\cdot \eta_{\mathrm{ku}}\cdot \eta_{\mathrm{pot}}} {i'-q_k}. \]

Fig. 8

Fig. 9

But

\[ i_k^{\prime\prime}=i_k'+\Delta i_k';\qquad \Delta i_k'=x\,\operatorname{tg}\beta_0 \quad\text{and}\quad y=x\cdot \operatorname{tg}\beta_1, \]

so that

\[ \Delta i_k' = y\cdot \operatorname{tg}\beta_0/\operatorname{tg}\beta_1, \]

and, in addition,

\[ \frac{y}{\Delta''i'} = \frac{\Delta''i'\cdot \operatorname{tg}\beta_1-\Delta'i'\cdot \operatorname{tg}\beta_2} {\Delta''i'\cdot \operatorname{tg}\beta_1}. \]

Taking approximately \(\beta_1 \simeq \beta_2\), we have \(y=\Delta''i'-\Delta'i'\), as in the preceding case, and, consequently,

\[ \Delta i_k' = (\Delta''i'-\Delta'i')\operatorname{tg}\beta_0/\operatorname{tg}\beta_1 = (\Delta''i'-\Delta'i')\alpha'. \]

Thus

\[ \eta_k^{\prime\prime} = \frac{[i'-\Delta'i'-i_k'-(\Delta''i'-\Delta'i')\alpha']\eta_{\mathrm{e}}^{\prime\prime}\cdot \eta_{\mathrm{ku}}\cdot \eta_{\mathrm{pot}}} {i'-q_k}. \]

Therefore

\[ \eta_m/\eta_k''-1= \frac{\eta_m(i'-q_k)} {\left[i'-\Delta'i'(1-\alpha')-\alpha'\Delta''i'-i_k'\right]\eta_E''\eta_{ku}\eta_{pot}} -1 \tag{22} \]

After all that has been said, the product

\[ M=\left(1-\frac{E_m''}{E}\right)\left(\frac{\eta_m}{\eta_k''}-1\right)= \]

\[ =\left\{\mu\left[ \frac{i'-\Delta'i'-t_{cond}} {i_n'+\alpha(\Delta''i'-\Delta'i')-t_{cond}} -1\right]-1\right\}\times \]

\[ \times\left\{1- \frac{\eta_m(i'-q_k)} {\eta_E''\left[(i'-i_k')-(1-\alpha')\Delta'i'-\alpha'\Delta''i'\right]\eta_{ku}\eta_{pot}} \right\} \tag{23} \]

But, as is seen from my preceding article \[4\], \(\Delta'i'\) and \(\Delta''i'\) are functions of the internal diameter of the steam-pipeline passage \(d\), whence, in order to obtain the optimum diameter \(d_{opt}\), one must set \(dM/d(d)=0\), from which \(d_{opt}\) is determined. From the above-mentioned article it is seen that

NOMOGRAM

for determining heat losses

Inside the nomogram: EXAMPLE.
Given: \(l_1=150\ \mathrm{m}\); \(q=1000\ \mathrm{kcal}/\mathrm{m}^2\mathrm{h}\); \(D_{sec}=50\ \mathrm{kg/sec}\); \(d=150\ \mathrm{mm}\).
Find: \(\Delta'i'=1.2\ \mathrm{kcal}/\mathrm{kg}\).
\[ \Delta'i'=\frac{\pi d l_1 q}{3600D_{sec}} \]

Fig. 10

\[ \Delta'i'=\frac{\pi d L_1 q_0}{3600D_{sec}}=f_1(d), \]

where \(L_1\) is the equivalent length of the steam pipeline in m, taking into account the additional heat losses through flanges and all the fittings of the steam pipeline; \(q_0\) is the hourly heat loss per \(\mathrm{m}^2\) of the internal surface of the steam pipeline, and \(D_{sec}\) is the quantity of steam, kg/sec, transported by the given steam pipeline. In Fig.

In [10] a nomogram is given for determining \(\Delta' i\) as a function of variable arguments. As for \(\Delta'' i\), from the same article,

\[ \Delta'' i=\psi \left[1-\left(1-\frac{\xi}{d^{5.25}}\right)\right]^{0.165}=f_2(d), \]

where \(\psi=0.00943\,P_0 v_0\), and

\[ \xi=\frac{0.00183\,L_2 D_{\text{sec}}^{2}}{P_0 \gamma_0}, \]

with \(P_0\) being the steam pressure in the boiler in \(\mathrm{kg/m^2}\); \(\gamma_0\) and \(V_0\) the density and specific volume of the steam at boiler pressure; and \(L_2\) the equivalent length of the steam line, including its local resistances. Fig. 11 shows a nomogram for determining \(\Delta'' i\) as a function of the varying arguments, greatly facilitating and accelerating the determination of this complex function.

Nomogram for determining heat losses \(\Delta i\)

Fig. 11

The first derivative \(dM/d(d)\) has a rather complex expression; solving it after setting it equal to zero, \(dM/d(d)=0\), in order to determine \(d_{\text{opt}}\), using ordinary methods, has so far proved troublesome and cumbersome. In seeking the corresponding solutions relating to the simpler case of purely condensing or purely back-pressure plants, we had to establish that the optimum passage diameter \(d_{\text{opt}}\) turns out to be so exaggerated that the steam velocities are thereby excessively underestimated. An investigation of this question, carried out in the article mentioned above, shows, however, that it is possible to reduce the passage diameters considerably without greatly lowering the efficiency of the steam line. Therefore, in the case under consideration of a plant with combined heat production, it is important for us to know not so much \(d_{\text{opt}}\) as the law of variation of \(\eta_{\text{steam}}\) as a function of \(d\).

Below, the table shows one particular example of such a change, when
\(W = 1655.57 \cdot 10^8\ \mathrm{cal/year}\); \(E = 15 \cdot 10^7\ \mathrm{kWh/year}\); \(P_0 = 33 \cdot 10^4\ \mathrm{kg/m^2}\); \(t_{\mathrm{per}} = 400^\circ\mathrm{C}\); \(P_n = 2.0 \cdot 10^4\ \mathrm{kg/m^2}\); \(L_1 = 120\ \mathrm{m}\); \(L_2 = 150\ \mathrm{m}\); \(q_0 = 800\ \mathrm{cal/m^2}\). For the diameters of passages according to OST within the limits \(d = 0.2\text{--}0.5\ \mathrm{m}\), we find from the nomograms (Figs. 10 and 11) the values \(\Delta'i'\) and \(\Delta''i'\), which makes it possible easily to calculate, for these values, the magnitudes of the products \(M\).

For the indicated particular example we have:

Table 1

\(d\)	\(\Delta'i'\)	\(\Delta''i'\)	\(M\)	\(\eta_{\mathrm{par}}\)
0.2	0.366	56.5	2.32	0.90
0.25	0.450	11.2	1.94	0.98
0.3	0.549	4.2	1.90	0.99
0.4	0.732	1.12	1.88	0.995
0.5	0.910	0.28	1.87	0.998

It is clear from the table that at \(d = 0.25\ \mathrm{m}\) the losses of the steam pipeline already amount to \(2\%\), which will occur if the steam velocity in the steam pipeline reaches \(70\text{--}80\ \mathrm{m/sec}\), as is often recommended. In the present case it would be more rational to stop at \(d = 0.3\ \mathrm{m}\), which gives more moderate steam velocities of \(50\text{--}55\ \mathrm{m/sec}\), while the losses of the steam pipeline will not exceed \(1\%\).

In analyzing the phenomenon of steam flow in a steam pipeline, we have so far encountered here one internal contradiction of the process, amounting to the fact that the requirement of smaller losses \(\Delta'i'\) to the environment forces us to reduce the diameter of the steam-pipeline passage, whereas the desire to reduce \(\Delta''i'\) requires increasing this diameter. But this is not the only contradiction in the phenomenon under investigation. The phenomenon cannot be considered in isolation from the surrounding circumstances, in the present case apart from connection with the general economy of the country, with which the steam pipeline is connected through the cost of its construction and the cost of a ton of steam, since fluctuations in the efficiency of the steam pipeline, affecting the specific fuel consumption in \(\mathrm{kg/kWh}\), may be admitted to a greater or lesser extent depending on fuel prices and the capital and production expenditures in the boiler house. Therefore it is inevitable to take into account also the influence of the economic factor in choosing the diameter of the steam-pipeline passages. It is only necessary to remember that the principal, decisive factor is nevertheless the attainment of the maximum efficiency of the steam pipeline, and that under planned management of the economy the economy of contiguous areas of the national economy must be brought up to the level that ensures results harmonizing with the scientific requirements of our time. There is nothing utopian in this; the revolutionary experience of our country shows that once a task that must be solved is clearly posed, it is solved by the efforts of the human collectives called upon to carry it out. Prof. Deneke dealt with the determination of the diameter of steam-pipeline passages with allowance for the economic factor, and the final formula at which he arrived has the form [5]

\[ d_{vsh}^{5} = \frac{ \dfrac{776\left(1 + P / D_{chas}\right)}{\gamma_m} \cdot \delta_p / d_{rkh} \cdot \Sigma \xi / l \left\{ \dfrac{24.53}{d_{dsh}} \cdot \dfrac{l}{\Sigma \xi} + 1 \right\} \cdot D_{chas}^{3} }{ \Phi \cdot \sigma + \dfrac{(l + l_0)\varepsilon_{tr} + (n + n_0)\varepsilon_v}{\dfrac{l}{m b}} } \]

This formula can be represented in the form \(d^5=A_1+B_1/d\), where

\[ A_1= \frac{ \frac{776(1+P/D_{\text{hr}})\,\delta_p}{\gamma^m\, d_{\text{r.h.}}\cdot \sum \xi/l\cdot D_{\text{hr}}^3} }{ \Phi_3+\frac{(l+l_0)\varepsilon_{\text{tr}}+(n+n_0)\varepsilon_v}{l} \Big/ (m\cdot b) } \]

\[ B_1=A_1\cdot \frac{24\cdot 5\%l}{\sum \xi}. \]

Thus, the least-expensive diameter \((d_{\text{le}})\) is determined as the intersection of the exponential curve \(y_1=d^5\), constructed once and for all, since it does not depend on any local conditions, and the hyperbola \(y_2=A_1+B_1/d\), which reflects in the coefficients \(A_1\) and \(B_1\) the influence of the given specific situation (Fig. 12). The asymptotes of the hyperbola are the ordinate axis and a straight line drawn from the abscissa axis at a distance \(A_1\). In the case under consideration of a steam pipeline with a bore diameter selected by maximum efficiency on the basis of the above considerations, this \(d_{\text{opt}}\) diameter (optimal or a smaller one close to it) is a definite quantity which, when plotted on the abscissa axis, makes it possible to obtain point \(C\) on the exponential curve \(y_1\), a point that may at the same time also be a point of the hyperbola \(y_2\). Let the ordinate of point \(C\) be \(y_1'\). In order that \(d_{\text{opt}}\) may also be \(d_{\text{le}}\), it is necessary that the straight line drawn parallel to the abscissa axis at a distance \(A_1\) be able to be the asymptote of the hyperbola \(y_2\) passing through point \(C\), i.e., that \(y_1' > A_1\). For given initial steam parameters and its total flow rate \(D_{\text{hr}}\), we can influence the value of \(A_1\) by changing the ratio

Fig. 12

\[ \frac{ \frac{(l+l_0)\varepsilon_{\text{tr}}+(n+n_0)\varepsilon_v}{l} }{ m\cdot b }, \]

i.e., by changing the number \(m\) of hours of use of the steam pipeline, \(b\)—the cost per kg of steam, and

\[ \frac{(l+l_0)\varepsilon_{\text{tr}}+(n+n_0)\varepsilon_v}{l} \]

the price of the organized steam pipeline per \(1\ \text{m}\) of its length and \(1\ \text{mm}\) of its diameter; moreover, as we see, the latter quantity depends substantially on the lengths \(l_0\) and \(n_0\) of the number of valves of the reserve lines of the steam pipeline. Let us also note that we can influence \(A_1\) by changing \(\sum \xi/l\), the total value of the resistance referred to a unit length of the steam pipeline. Thus, not only the cost of the products of machine-building and metallurgical plants and the cost of installing the steam pipeline, but also the design of the steam pipeline affect the value of \(A_1\), which in the final analysis can be obtained as required in order to satisfy the condition \(y_1' > A_1\).

In reducing the diameter of the steam-pipeline passage in the name of economy, we are nevertheless obliged to give a clear account of the price, in terms of the reduction in the efficiency of the steam pipeline and, consequently, the overconsumption of fuel, that we pay for this, in order to make clear to the adjacent branches of the economy what sacrifices they force upon us by not bringing production up to the requirements of heat engineering.

Other Formulas for Efficiency and Their Critique

Without multiplying further the number of examples of application of our method, let us now turn to the briefest review of other formulas for the efficiency of a CHP plant, proposed by other authors. Prof. Dollezhal’ [6], starting from the same initial expression as we do,

\[ \eta=\frac{N\,860^{\mathrm{netto}}+Q_{\mathrm{use}}}{Q_{\mathrm{fuel}}}, \]

for a back-pressure installation obtained

\[ \eta = \frac{860\,N_{\mathrm{e}}}{Q_1}\cdot \frac{Q_1}{Q_{\mathrm{fuel}}}\cdot \frac{N^{\mathrm{netto}}}{N_{\mathrm{e}}} + \frac{Q_{\mathrm{use}}}{Q_1}\cdot \frac{Q_1}{Q_{\mathrm{fuel}}} = \]

\[ = \frac{Q_1}{Q_{\mathrm{fuel}}} \left[ \eta_{\mathrm{el}}\,(1-K_{\mathrm{e}}) + \frac{Q_{\mathrm{use}}}{Q_1} \right], \]

where

\[ K_{\mathrm{e}}=1-\frac{N^{\mathrm{netto}}}{N_{\mathrm{e}}} \]

and \(Q_1=D_1 i_1\) is the heat of the steam before the turbine.

But

\[ \frac{Q_{\mathrm{use}}}{Q_1} = \frac{D_u i_u}{D_1 i_1} = K_u i_u i_1 = K_u\frac{i_1-h_a\,a\,\eta_{\mathrm{ua}}}{i_1} = K_u\left(1-\frac{h_a}{i_1}\,a\,\eta_{\mathrm{ua}}\right) = \]

\[ = K_u(1-\eta_{\mathrm{ua}}\,a), \]

moreover

\[ \frac{Q_1}{Q_{\mathrm{fuel}}} = \frac{D_1 i_1\eta_{\mathrm{ku}}}{D_{\mathrm{pe}}(i_{\mathrm{pe}}-i_{\mathrm{fw}})} = \frac{D_1 i_1\cdot i_{\mathrm{pe}}\eta_{\mathrm{ku}}} {i_{\mathrm{pe}}\cdot D_{\mathrm{pe}}(i_{\mathrm{pe}}-i_{\mathrm{fw}})} = \]

\[ = (1-K_p)\,\eta_{\mathrm{par}}\,\eta_{\mathrm{ku}}\cdot \frac{i_{\mathrm{pe}}}{i_{\mathrm{pe}}-i_{\mathrm{fw}}} = (1-K_p)\eta_{\mathrm{par}}\eta_{\mathrm{ku}}K_r, \]

where

\[ K_p=\frac{D_{\mathrm{own}}}{D_{\mathrm{pe}}} \]

is the coefficient of steam self-consumption, and

\[ K_r=\frac{i_{\mathrm{pe}}}{i_{\mathrm{pe}}-i_{\mathrm{fw}}} \]

is the regeneration coefficient, giving an idea of the degree of use of extraction. Therefore, finally,

\[ \eta = \eta_{\mathrm{ku}}\cdot\eta_{\mathrm{par}}K_r(1-K_p) \left[ \eta_{\mathrm{el}}\cdot(1-K_{\mathrm{e}}) + K_u(1-a\eta_{\mathrm{ua}}) \right] \tag{24} \]

This formula (24) has the following shortcomings:

1) it does not cover the independent boiler-house and purely condensing part of the installation or system, which, as we have seen, is a substantial defect;

2) the back-pressure part may have not one steam extraction, but more, with different pressures for external consumers, which is not reflected in the formula;

3) the formula does not take into account the influence of the number of steam-extraction points for regeneration: it follows that the efficiency will be the same whether we take one steam-extraction point, heating the water at once to \(t_{\mathrm{fw}}=i_{\mathrm{fw}}\), or take several steam extractions, heating the feedwater in stages, which should ensure a closer approximation to the efficiency of the Carnot cycle;

4) the formula does not take into account the loss of heat in full during its transport within the station;

5) electrical and mechanical losses are charged to electricity alone.

In addition to all that has been said, an incorrect interpretation is given of the influence of the absolute efficiency \(\eta_{ka}\), approximately in the same spirit as in Prof. Hasen, without taking into account the influence of the condensing part of the installation. Thus, Prof. Dollezhal’s formula, in the form in which it is given, cannot count on practical application.

Prof. Luknitskii, V. V. [7], comes very close to Prof. Dollezhal’s point of view; he proceeds from the formula for the net efficiency of an installation with back pressure and complete utilization of the exhaust heat

\[ \eta_{cm}^{netto}= \frac{860\,E_{evp}(1-K_e)+Q}{B\cdot Q_p^n}, \]

where \(Q=D_n\cdot i_n\) is the quantity of heat delivered to consumers. After transformations analogous to those indicated in deriving Prof. Dollezhal’s formula, this formula gives the net efficiency of the station in the form

\[ \eta_{cm}^{netto}=\eta_{kv}\cdot \eta_{par}\cdot \alpha(1-K_e) \left[ \eta_e(1-K_e)+\frac{D_n\cdot i_n}{D_o\cdot i_o} \right] \tag{25} \]

Since for an engine with final utilization of heat \(D_n=D_o\), the absolute net efficiency of this engine is obtained as \(\eta_{dv}=\eta_e(1-K_e)+\dfrac{i_n}{i_o}\), or

\[ \eta_{dv}=\eta_e(1-K_e)+\frac{i_o-H\eta_{oi}}{i_o} = 1-\eta_k\eta_{oi} \left[ 1-\eta_{gen}\cdot \eta_{mech}(1-K_e) \right] \tag{26} \]

From this Prof. Luknitskii concludes that the total net efficiency of an engine with final utilization of heat will be the smaller, the more perfect the internal process in the engine and the greater the available heat drop, i.e., the higher the initial steam parameters and the lower the back pressure. This conclusion appears so unexpected that Prof. Luknitskii considers it necessary on the following page to develop it once again, while indicating that the maximum net efficiency of the station is obtained when throttled steam from the boiler is used, for then \(\eta_k\) and \(\eta_{oi}\) are equal to zero. This maximum will then be equal to the efficiency of the boiler plant. All this is the same as in Prof. Hasen.

Prof. Luknitskii charged electrical and mechanical losses exclusively to electricity, which, as indicated above, cannot be regarded as correct. Further, the author charged the station’s own-use electricity consumption to the engine, whereas the back-pressure engine, as such, has no such consumption, since it has no circulating and condensate pumps; still less can the consumption for the own needs of the boiler plant and of the station as a whole be charged here.

The remarks made concerning Prof. Dollezhal’s formula also apply in considerable part to the case under consideration. In concluding the examination of the two proposals described, it must be said that their greatest shortcoming is the refusal to pose the question more broadly, motivated in Prof. Luknitskii by its great complexity, which in fact, as we have seen from the preceding discussion, does not exist. The absence of broad averaging, as a rule, leads to erroneous conclusions.

Taking a negative attitude toward formula (1), Prof. G. I. Fuks [8] proposes replacing it by a “degree of perfection.”

The essence of this new proposal reduces to the following: the degree of perfection

\[ \mu=\frac{E_{\text{obt}}}{E_{\max}}, \]

where \(E_{\text{obt}}\) and \(E_{\max}\) are the amount of energy actually obtained from the installation and the maximum possible amount of obtaining it according to the laws of nature. In particular, in the case of a combined heat-and-power plant,

\[ \mu=\frac{E_{\text{obt}}}{E_{\max}} = \frac{860N+Q_r(1-T_0/T_{cm})}{B\cdot \mathfrak{A}_0} \tag{27} \]

Here: \(\mathfrak{A}_0\) is the maximum work of the combustion reaction of a unit of fuel under ambient conditions; \(B\) is the hourly fuel consumption; \(Q_r\) is the hourly heat consumption by heat consumers; \(T_0\) is the absolute temperature of the surrounding medium, into which the spent heat is discharged and from which heat energy can be drawn without limit; \(T_{cm}\) is the mean thermodynamic temperature of the heat process of heating the working body of the installation,

\[ T_{cm}=\frac{q_m}{\Delta S_m}, \]

where \(q_m\) is the heat of this process, and \(\Delta S_m\) is the change in entropy of the working body in this process. It is assumed here that the work obtained in reversible combustion of the fuel is expended on the generation of electric energy and on actuating a heat pump, which pumps heat from the surrounding medium to the heated body of our installation. At the expense of \(\mathfrak{A}_0\) cal/kg of work developed by 1 kg of fuel, there is obtained

\[ U_m=\frac{\mathfrak{A}_0}{AW} \]

kg of heated body, so that

\[ E_{\max}=\frac{\mathfrak{A}_0}{AW}\cdot q_\tau=U_m\cdot q_\tau . \]

If in reality \(U_\tau\) kg is heated, then heat

\[ E_{\text{util}}=U_\tau\cdot q_\tau \]

is utilized, so that

\[ \psi=\frac{U_\tau}{U_m} = \frac{U_\tau\cdot A\cdot W}{U_m\cdot A\cdot W}. \]

But from the \(TS\) diagram we have (Fig. 13)

\[ AW=q_\tau-T_0\cdot \Delta S_m, \]

whence

\[ \psi = \frac{U_\tau(q_\tau-T_0\Delta S_\tau)}{\mathfrak{A}_0} = \frac{U_\tau q_\tau}{\mathfrak{A}_0}(1-T_0/T_{cm}). \]

Fig. 13

Passing to hourly quantities, we obtain

\[ \mu= \frac{860N+Q_r(1-T_0/T_{cm})}{B\cdot \mathfrak{A}_0} \]

The proposal described requires the realization, in the working body of the installation, of high temperatures close to furnace temperatures. Neither in steam boilers nor in turbines can such temperatures yet be achieved, owing to the absence of the necessary materials. Proposals to evaluate the degree of perfection of boiler installations with allowance for the increase of entropy occurring in them were made as early as the beginning of the twentieth century by a number of authors (Gouy, Stodola, Krauss, and others), so that G. I. Fuchs is not a pioneer here; but all these proposals were not accepted in scientific and engineering circles, one must think, for the same reason for which they pass over in silence, vis-

for the unused hydropower of rivers and waterfalls somewhere in the Altai. At the present stage of our development we are not in a position to master completely either the thermal head in steam boilers or the head of the majority of our rivers. This impossibility of mastering the thermal head in boilers is also due to the inevitable increase, in doing so, of heating surfaces and of metal consumption, which at present would not be economically justified. Therefore our attention is naturally directed toward the best utilization of heat at realizable heads in boilers, being focused on the struggle to preserve every calorie and to reduce the amount of metal in the structure, and on the possible increase of steam parameters. We simply have no practical need at the present moment for any other evaluation than by the generally accepted efficiency. That is why we do not now take into account the loss of thermal head in boilers, which could theoretically provide additional generation of electric power.

Great difficulties are presented by the choice of the magnitude \(T_0\). The author believes that, in determining the degree of perfection of a heat-power installation, \(T_0\) should be understood as the absolute temperature of the surrounding air. But here a difficulty arises: heat-power installations operate at different air temperatures. Therefore the author considers it the most tempting course, for the possibility of comparing installations, to determine \(T_0\) at some conventional “normal” ambient temperature. Analyzing statistical data for average annual air temperatures over a considerable part of the USSR, the author comes to the conclusion that \(T_0 = 273\ \mathrm{K}\). But the author emphasizes that \(T_0 = 273\ \mathrm{K}\) may be adopted only when it is necessary to compare installations operating under different conditions. In studying a given installation, however, \(T_0\) should not be assumed, but measured directly from experiment. Thus the author has no definiteness in the choice of \(T_0\): two scales are proposed, as it were, which, of course, complicates the situation. The presence of sharp temperature fluctuations across the latitude of the USSR and by seasons of the year may compel the division of the country into zones with different temperatures, which will make the complexity of the evaluation still worse.

The process of an engine installation—heat pump—considered by Prof. G. I. Fuchs differs essentially from the real process at a heat-and-power plant, where the heat carrier is heated directly by exhaust steam, and therefore it cannot be related to the latter process. Transferring the regularities of the first process to the second is inadmissible and may lead to serious discrepancies. Suppose, for example, that a heat-and-power plant supplies extraction steam of identical parameters only for technological purposes in equal quantities in July \((t_0 = +20^\circ\mathrm{C})\) and in January \((t_0 = -20^\circ\mathrm{C})\), with the same generation of electric power. According to formula (2), the efficiency of the heat-and-power plant will be the same both in January and in July, as will the fuel consumption. According to Prof. G. I. Fuchs’s method, however, since \(T_0/T_{st}\) will be greater in July than in January, the fuel consumption in July will prove to be less, whereas this consumption will in fact be the same at the heat-and-power plant in both of the indicated months. It is a different matter if the production of heat is provided by a heat pump; then in July the fuel consumption will indeed be lower, since the degree of compression is also lower. With direct heating of the heat carrier, however, the difference in fuel consumptions in the particular case considered cannot be explained in any way.

In No. 4 for 1948 of the journal Bulletin of Engineers and Technicians there appeared an article by Prof. D. P. Gokhshtein, “Application of the Second Law of Thermodynamics to the Analysis and Calculation of District Heating.” In this article the author asserts that “the useful work of chemical and physico-chemical processes has the same energetic value as the useful work of an engine.” As far as can be understood, such an assertion is a conclusion by analogy, like the assertion that “just as energetically

valuable is that part of the heat used for heating which could have been converted into maximum work under definite parameters of the power cycle.” The correct conclusion by analogy must satisfy, in the opinion of logicians, certain conditions, the chief of which is that the similarity must predominate over the difference. If Figs. 3, 4, 5 of the article are regarded as power cycles, then all of them use thermal drops within the limits from the pressure corresponding to the temperature \(T\) of the heating steam to the pressure at the temperature \(T_0\) of the environment. This is the normal form of the power cycles of a theoretical condensing engine, in which no use is made of the latent heat of vaporization of the exhaust steam. If, for simplicity, we speak of the Carnot cycle (Fig. 14), then the latter, at temperature \(T\), will be represented by the rectangle \(adefa\). This heat of vaporization will be transferred to heat consumers without, of course, obtaining any work. Prof. Gokhshtein asserts that “the summation of incommensurable terms—exhaust heat and mechanical work—and hence the view of them as equivalent components of the utilized part of the heat of the fuel, is a logical consequence of a gross confusion of the statements of the first and second laws of thermodynamics.” Meanwhile, every heat engineer knows that the area \(adefa\) represents the work performed in the temperature interval \(T\) and \(T=0\). Consequently, there is nothing contradictory when in the numerator a summation is made of two commensurable terms, just as no one finds it contradictory when, in determining the efficiency of a purely power cycle, the areas \(abcda\) and \(cbfec\) are added in the denominator. Here it is only necessary to bear in mind that the basement of the cycle represents, on the one hand, the latent heat of vaporization at temperature \(T\), used by the heat consumers, and, on the other hand, the possible work in the temperature interval \(T\) and \(T=0\).

Fig. 14

The difficulty of recognizing the correctness of the analogy used may also be seen from the following. If it is assumed that in winter at \(-40^\circ\text{C}\) chemical and physicochemical processes will have lower values of \(T_0\), then in the power process, in its back-pressure part, the temperature \(T\) of steam extraction remains unchanged, just as the latent heat of vaporization at this temperature will also remain unchanged. Therefore the efficiency of the power cycle in the part under consideration remains unchanged in winter and in summer, whereas in Prof. Gokhshtein’s interpretation the power “technical” cycle will have a higher efficiency in winter, and in summer, at temperature \(t=+40^\circ\text{C}\), it will be lower. From what has been said it is clear that in Prof. Gokhshtein’s reasoning we have an example of the paralogism petitio principii, when the author used as a premise a judgment which remains disputed, while the author himself treats it as though it were indisputable.

From everything said in the present work it follows that the meaning of the application of district heating consists chiefly in the fact that precisely in this way it becomes possible to utilize the latent heat of vaporization of the exhaust steam, which in purely condensing installations is carried away with the cooling water of the condenser. The considerations concerning the degradation of energy which Prof. Gokhshtein advances as the principal criterion of district heating-

...have only a subordinate significance when one has to discuss the advantages of extracting steam from the engine as compared with supplying it from the boiler, or when the choice of pressure at the steam-extraction points is at issue. All these questions are quite fully resolved without applying Prof. Gokhshtein’s reasoning, proceeding, as we have seen, from the heat balance of the CHP plant and from our formulas.

Among the opponents of formula (1) is also A. S. Gorshkov [9], who believes that no losses should be charged to the heat energy supplied by a back-pressure unit, since they are connected exclusively with the production of electric energy, and therefore the efficiency for heat production here is \(\eta^T = 1\); whereas the efficiency of electric-power production is

\[ \eta_{\mathrm{e}}=\frac{860\,E}{Q_{\mathrm{e}}}, \]

where \(Q_{\mathrm{e}} = Q_n - Q_k - Q^T\), and \(Q_n\) is the supplied heat, \(Q_k\) is the heat of the condensate sent to the boiler house, and \(Q^T\) is the heat of external consumption. We, however, take \(\eta^T = \eta'_{\mathrm{mech}}\cdot \eta'_{\mathrm{gen}}\). Mechanical and electrical losses will, to a certain extent, necessarily affect the heat consumers as well. If these losses did not exist, then for a given magnitude of external power and given initial and final steam parameters we would require an internal power of the unit \(N'_{\mathrm{in}}\), to which a steam flow \(D^1\) would correspond. Because of the presence of mechanical and electrical losses, however, it is necessary to generate an additional power inside the unit \(N''_{\mathrm{in}}\), for which an additional steam flow \(D_2\) will be required; moreover, the additional heat consumption will be determined as \(D_2(i_1-i'_n)\eta^i\) cal. This heat loss, being taken from the heat flow, is allocated between the electric and heat consumers, as formula (3) indicates; it cannot fail to affect the heat consumers. This is especially clear from the heat-balance diagram (Fig. 15). Having overlooked this, A. S. Gorshkov naturally came to a rejection of the method of “equality of energy values,” i.e., of formula (1).

In contrast to the investigators who sought to evaluate the CHP-plant process by a single efficiency, A. S. Gorshkov, and after him the entire Ministry of Power Stations, plans and accounts for the techno-economic indicators separately for electric energy and separately for heat energy; moreover, when calculating the fuel consumption for the heat energy supplied, the efficiency of heat production is taken equal to the efficiency of the boiler installation. In the fuel consumption for the production of electric power, besides the boiler efficiency, all other losses are taken into account, including the mechanical and electrical losses of the turbogenerator. This is justified by saying that the techno-economic indicators of a CHP plant must characterize the peculiarities of the production of each type of energy. As early as Herzen [10] wrote: “Life is the preserved unity of diversity, the unity of the whole and of the parts.” Herzen gives a fine illustration of this: “Every branch of a tree, even every bud, has its own relative individuality; they may be taken as separate plants; but the totality of them belongs to one whole, the living plant of these plants—the tree; remove the branches and a dead stump remains, remove the trunk and the branches fall apart.” Our formulas (5, 10, 12) provide this connection.

Fig. 15
(Labels in the diagram: heat consumers; losses of the condensing unit; losses of the water piping; electrical and mechanical losses; production of electric energy; heat of condensate; fuel \(W\); fuel consumers.)

between the efficiency of the integral process of a CHP plant (binary installations, the power system) and the partial efficiencies relating to the “branches” of this whole. Ignoring the whole, and broad mediation, must inevitably lead to errors in theory and, as a result, to incorrect policy on the energy front, to distortions in practice. This is just the case in the Ministry of Electric Power Stations, for the adoption of \(\eta_m=\eta_{ky}\) instead of

\[ \eta_m=\eta_{ky}\cdot\eta_{\mathrm{pot}}\cdot\eta'_{\mathrm{mek}}\cdot\eta'_{\mathrm{gen}}, \]

when

\[ \eta_{ky}>\eta_{ky}\cdot\eta_{\mathrm{pot}}\cdot\eta'_{\mathrm{mek}}\cdot\eta'_{\mathrm{gen}}, \]

understates the fuel consumption per 1 megacalorie, increasing the specific fuel consumption per 1 kWh for considerations that have nothing in common with the actual process at the CHP plant. The indicators of the specific consumption of standard fuel in the Ministry of Electric Power Stations system are not a correct reflection of reality.

It is interesting to note in passing that A. S. Gorshkov, who declared himself an implacable enemy of formulas of type (5), where the expression

\[ \frac{860\mathcal{E}+W}{\eta} \]

figures (Gorshkov, Op. cit., p. 85), is in fact captive to this formula. Indeed, from our formula (4) it follows that

\[ Q_k=Q_k+Q_m=\left[\frac{860(\mathcal{E}-\mathcal{E}_m)}{\eta_k}+\frac{860\mathcal{E}_m}{\eta_m}\right]+\frac{W}{\eta_m} =\frac{860\mathcal{E}+W}{\eta}, \]

i.e., here we have techno-economic indicators in two terms, separately for electric energy in square brackets and separately for heat energy; both these terms are connected by the common quantity

\[ \frac{860\mathcal{E}+W}{\eta}, \]

as their sum. The disagreement with A. S. Gorshkov, as indicated above, consists only in the value of the efficiency \(\eta_m\). Thus A. S. Gorshkov only outwardly, as it were, emancipated himself from the general averaging efficiency \(\eta\) of the CHP plant, but for this emancipation he paid the high price of losing an understanding of the integral process at the CHP plant,^1 which inevitably led him to exaggerate one side of the whole, directing him along the path of an idealistic interpretation of the CHP process. A. S. Gorshkov did not understand the internal dialectic of the process, just as B. M. Yakub [11] did not understand it either, who gave an isolated formula for the efficiency of electric-power generation at a CHP plant:

\[ \eta=\frac{\eta_w\cdot\eta_{wq}}{\eta_{wq}-\dfrac{\mathcal{E}_q}{\mathcal{E}}(\eta_{wq}-\eta_w)} \tag{28} \]

Here \(\eta_w\) is the efficiency of electric-power generation by the condensing method; \(\eta_{wq}\) is the same for heat consumption, \(=\eta_{ky}\cdot\eta'\eta_{\mathrm{mek}}\cdot\eta_{\mathrm{gen}}\), where \(\eta'\) is the efficiency of the steam pipeline; \(\mathcal{E}_q\) is the generation of electric power in kWh for heat consumption. This formula (28) is obtained from the initial equation

\[ \frac{\mathcal{E}-\mathcal{E}_q}{\eta_w} + \frac{\mathcal{E}_q}{\eta_{wq}} = \frac{\mathcal{E}}{\eta}. \]

B. M. Yakub’s formula (28) differs from our formula (5) in that it ignores the heat delivered to heat consumers and, consequently—

^1 Defending his positions in No. 6 for 1948 of the journal Electric Stations, engineer Gorshkov criticized what he alleged to be my formulations, and moreover constructed his objections while polemicizing against the old formula of Prof. Grinevetsky

\[ \eta=\frac{860\mathcal{E}+W}{Q}, \]

the significance of which he denied; as indicated above, as early as 1939 A. S. Gorshkov did not want to understand that my proposals unite Grinevetsky’s formula with Gorshkov’s partial efficiencies in a higher synthesis: that is, the negation of engineer Gorshkov’s negation. Thus all his objections missed the mark.

...accordingly, seeks to take into account only electric power as the sole product of a CHP plant. Thus, this is not the absolute efficiency of a CHP plant, and B. M. Yakub’s formula cannot in any way serve to determine the fuel consumption of a CHP plant. B. M. Yakub’s formula (29), moreover, takes into account in heat transport only the thermal loss of the steam pipeline, while the mechanical and electrical losses are fully charged in it to electric power; the incorrectness of this was indicated above.

V. V. Tumasov [12], by the “efficiency of heat production,” determines the share of the fuel burned at the CHP plant that is attributable to heat consumers, according to the formula

\[ B_T=\frac{Q\cdot 10^3 Q}{\eta_T \cdot Q_p^h}\, m, \tag{29} \]

where \(Q=\dfrac{W}{10^6}\), and \(\eta_T=\eta_{ku}\cdot \eta' \cdot \eta_{sp}\cdot \eta_{tu}\). Here \(\eta_{sp}=\dfrac{D_{iz}}{D_{kot}}\) is a coefficient characterizing steam losses and its self-consumption for the plant’s own needs; \(\eta'\) is the efficiency of the steam pipeline; \(\eta_{tu}\) is the efficiency of boilers and steam converters.

Earlier, in accordance with reality, we established that it is necessary to distinguish between the production of heat from engine extractions and directly from boilers, and to take them into account separately. From this point of view it is incomprehensible what relation \(\eta_{sp}=\dfrac{D_{iz}}{D_{kot}}\) can have to \(Q=\dfrac{W}{10^6}\) of the heat energy produced from extractions, measured in million kcal, and to the fuel consumption for heat production

\[ B_m=\frac{Q\cdot 10^3}{\eta_T\cdot Q_p^h}, \]

since \(Q\) depends exclusively on the consumption of heat by heat subscribers, while the coefficient \(\eta_T\) must characterize losses in the process of obtaining this heat; moreover, consequently, the losses of the boiler installation and of heat transport within the station must be taken into account, and not only those of the steam pipeline, as in V. V. Tumasov; and, as indicated above, for heat from extractions, that part of the electrical and mechanical losses of the turbogenerator which falls to the share of the heat subscribers. Therefore \(\eta_T=\eta_{ku}\cdot \eta_{pot}\cdot \eta'_{mekh}\cdot \eta'_{gen}\), and for live steam \(\eta_T=\eta'_{ku}\cdot \eta'_{pot}\). V. V. Tumasov also artificially isolated “heat production,” excluding it from the integral process of the CHP plant, and therefore there is at present no reason to use his proposal.

Thus, the common shortcoming of the three proposals described (A. S. Gorshkov, B. M. Yakub, and V. V. Tumasov) is that they separated the “branches” from the “trunk.” Their one-sidedness was a reaction to the one-sided use of the “trunk”

\[ \frac{860\,\mathcal{E}+W}{\eta} \]

without “branches,” which in turn led to an exaggeration of the other side, torn away from the actual process at the CHP plant, and entailed errors in practice. The correct solution lies in combining both sides of a single integral process, which our method provides.

Having rejected the original formula (2) for a back-pressure installation,

\[ \eta=\frac{860\cdot \mathcal{E}_m+W}{Q_{topl.}}, \]

on the unfounded grounds set forth above, Prof. Khazen naturally turned to finding a way out of the situation he himself had created and came forward with a new proposal (13), consisting in comparing the operation of any real installation with a certain reference installation at a specified heat production and heat consumption. The Rankine cycle is adopted as the reference one, in which, for water vapors, \(p_1^{ref}=224.2\ \text{ata}\); \(t_1^{ref}=550^\circ\text{C}\); \(p_2^{ref}=0.03\ \text{ata}\), for

of mercury vapor: \(p_1^{\mathrm{obr}} = 13.6\ \mathrm{ata}\), \(t_1^{\mathrm{obr}} = 550^\circ\mathrm{C}\); \(p_2^{\mathrm{obr}} = 0.1\ \mathrm{ata}\). We saw above that, in essence, there was no need to introduce, for the assessment of the perfection of a CHP plant, some new method, a series of new unnecessary concepts, and to complicate this simple and clear matter. Independently of this, it is unclear why precisely the steam parameters just indicated are adopted as if they were limiting, as if ideal, toward which we should strive. Robb, for example, as early as 1925 envisaged, for a pressure of the order of \(200\ \mathrm{ata}\), a steam temperature of \(600^\circ\mathrm{C}\). It is also unclear why, for mercury, \(p_1^{\mathrm{obr}} = 13.6\ \mathrm{ata}\) and \(p_2^{\mathrm{obr}} = 0.1\ \mathrm{ata}\), if one takes into account not only Robb’s indicated proposal and the considerations set forth above concerning the choice of the optimum back pressure in a mercury turbine, but also the necessity of resorting to thermochemical cycles in the steam-water part of the plant in order to avoid an increase in moisture at the end of the expansion of the mercury vapor when passing to higher parameters of it in the mercury boiler. There is no doubt that at \(p_1^{\mathrm{obr}} = 224.2\ \mathrm{ata}\) and at \(t_1^{\mathrm{obr}} = 550^\circ\mathrm{C}\) secondary superheating of the steam will be required, which, as is known, will raise the efficiency of the recommended ideal cycle. Prof. Yanovskii [14] justly notes a number of other sufficiently large discrepancies in Prof. Khazen’s article.

Also of interest are the critical remarks made with respect to the basic propositions of Prof. Khazen in the article by Candidate of Technical Sciences V. T. Yurinskii [15].

The Cost of Energy Products at a CHP Plant

Let us first make a brief review of previously published proposals for determining the cost of energy products at a CHP plant, with criticism of them and an indication of the reasons for their inapplicability in practice. A number of authors (Arons, Ryzhkin, and others) proposed that the cost of a ton of steam used for heating needs be calculated on the basis of the value of the underproduction of electric power caused by extraction of steam, since the latter does not receive, in the combined plant, the possibility of full expansion to condenser pressure. The cost of extraction steam is here taken to be proportional to the cost of live steam and to the quantity \(\frac{h_k - h'}{h_k}\), where \(h_k\) is the adiabatic drop in the engine from the initial pressure to the pressure in the condenser, and \(h'\) is likewise the drop to the extraction pressure [16]. These proposals are based, consequently, on the point of view that value is possessed only by steam within the limits of the full heat drop used in the operation of the engine to the condenser. This point of view in essence reflects a stage already passed through in the development of power stations, when they were exclusively condensing stations, and the exhaust steam going to the condenser, not being an object of consumption, could have no cost. It is clear that the cost of heat used for heating purposes is obtained by this method as obviously understated, since the latent heat of the extraction steam is not taken into account.

Another group of proposals, on the contrary, leads to an exaggerated cost of heat for the latter purpose. This includes, for example, the method of determining the cost of steam going for heating purposes by multiplying the cost of live steam by the ratio of the heat contents of extraction steam \(i_n\) and of live steam \(i\). In view of the large value of the fraction \(i_n/i\), the cost of extraction steam is obtained here as differing little from the cost of live steam. In the historical development of the principle of the combined production of heat there was a moment when many began to regard the electric power obtained in the combined process as a by-product, as though a production waste. It is therefore natural that its cost was taken as understated, and the heat-content method corresponded precisely to this view.

When the extremes of both groups of proposals indicated above had become sufficiently clear, when it had become evident that electricity and heat satisfy equally important needs of the national economy and of everyday life, attempts were made to solve the problem by compromise solutions. Among these may be counted, for example, the proposal of engineer Svechin to take the coefficient of reduction in the cost of extraction steam as compared with live steam not as \(i_n/i\), but as \((i_n/i)^2\). To the same category belongs the proposal of engineer Rumyantsev, recommended even for practical use by the “Provisional Instructions for Drawing Up Estimates of an Energy Center” (1931), and at present no longer used in the MES system, but not yet out of use at Transtechproekt of the Ministry of Railways and in the design organizations of certain industrial ministries. In Rumyantsev’s method the coefficient of reduction in the cost of extraction steam as compared with live steam has the form [17]

\[ \frac{ i_n/i+\dfrac{h_k-h_i}{h_k} }{2} \]

There is no need to say that all such proposals, containing elements of conventionality and of a known arbitrariness, cannot in principle correctly reflect reality: by their very nature they are unscientific, invented out of thin air, and must be decisively rejected as obviously idealistic. In all the methods described, the cost of 1 kWh can be determined, using the well-known Ginter triangle, after the cost of a ton of extraction steam has been determined.

The basis of engineer Rumyantsev’s proposal is the premise that “the only practically expedient and logically acceptable method is the distribution of the saving (from the combined production of heat and electricity) uniformly over both kinds of energy, making them cheaper in equal proportion,” since “there is no doubt that both consumers have the right to demand equal benefits for themselves.” Prof. V. V. Luknitskii [18], recommending the methods of Rumyantsev and Pakshver, writes that “other methods do not meet the needs of life, since they give unacceptable and sometimes absurd results and may have a retarding influence on the spread of the combined method of producing heat and electricity.” From these two quotations it is clear that their authors stand on an anthropomorphic point of view, taking care of “logical acceptability,” “satisfaction of the needs of life,” “practical expediency,” that there should be no “retarding influence on the spread of the combined method of producing heat and electricity,” and also that consumers have the right to demand equal benefits for themselves. They are completely uninterested in the process of combined energy production in the form in which it actually takes place at a CHP plant, independent of our human evaluations. The method of engineer Pakshver mentioned above, for example, which “should be applied” (Prof. Luknitskii), consists in first determining the price of steam and electricity on the assumption that they are produced separately: steam—in a boiler house, and electricity—at a condensing station. From the total annual costs of the CHP plant are subtracted the costs falling on heat subscribers, obtained by multiplying the costs of the live steam supplied to them by a fractional coefficient determined using Ryzhkin’s method. The difference is taken as the cost of producing electricity at a condensing power station. Finally, with the aid of the Ginter triangle, the cost of the energy products at the CHP plant is determined, assuming that this cost will be proportional to the values of cost obtained at the mentioned separate installations.

From what has been said, the entire contrived artificiality of Eng. Pakshver’s constructions, which have nothing in common with the real process at a CHP plant, is evident.

More recently, proposals have appeared by Eng. Andryushchenko [19], who proceeds from a principle analogous to that of Eng. Rumyantsev, namely that “thermal and electrical consumers must receive equal economy from combined generation.” The author’s reasoning is as follows. The cost of 1 kg of live steam is \(A' = z/D\), and the cost of the heat in 1 kg of live steam is \(A = A' - C_k\), where \(C_k\) is the cost of the condensate. In the case of supplying steam to a heat consumer directly from the boiler, the cost of 1 useful cal, provided that condensate at temperature \(t_k\) is returned, will be

\[ \beta = \frac{A}{i_1 - t_k}. \]

In the case, however, of generating electric power by the condensing method, the cost of one cal converted into electric power is

\[ \alpha = \frac{A}{i_1 - i_{2k}}, \]

where \(i_{2k}\) is the heat content of the steam downstream of the turbine. In the case of combined generation, let us denote the cost of 1 cal converted into electric power by \(x\), and into heat energy by \(y\). Proceeding from the principle that thermal and electrical consumers must receive equal economy, we write \(x/y = \alpha/\beta\), i.e.

\[ x/y = \alpha/\beta = \frac{A(i_1 - t_k)}{A(i_1 - i_{2k})} = \frac{i_1 - t_k}{i_1 - i_{2k}} \tag{30} \]

On the other hand, the cost of the heat of 1 kg of live steam is

\[ A = x(i_1 - i_2) + y(i_2 - t_k) \tag{31} \]

Solving (30) and (31), we obtain

\[ x = \frac{A(i_1 - t_k)} {(i_1 - i_{2k}) \left[ \frac{(i_1 - t_k)(i_1 - i_2)}{i_1 - i_{2k}} + i_2 - t_k \right]} \ \frac{\text{kop}}{\text{cal}}, \quad \text{and} \]

\[ y = \frac{A} { \frac{(i_1 - t_k)(i_1 - i_2)}{i_1 - i_{2k}} + i_2 - t_k } \ \frac{\text{kop}}{\text{cal}} \]

The principle that “the heat and the electrical consumer must receive equal economy from combined generation” is as arbitrary as any other principle of allocation, and in its essence differs in no way from the analogous proposals of Rumyantsev, Svechin, and others. This proposition does not reflect reality, but is invented out of whole cloth. It is not an axiom, but a proposition that itself requires proof, all the more so because proportionality itself is adopted with respect to the costs of 1 cal received by heat subscribers directly from the boiler, and 1 cal converted into electric power under condensing operation, i.e., with respect to the costs of calories whose production at the CHP plant occurs under entirely different conditions. There are no grounds for thinking that Eng. Andryushchenko’s method could count on general recognition, being a relapse into essentially the same idealistic constructions.

Let us now turn to the new formula of B. M. Yakub [20]. The “no-load” expenditures \(B\) (depreciation, personnel maintenance) are taken to be proportional to the installed capacity \(N_y\) of the CHP plant and to the maximum heat delivery \(Q_y\) from the CHP plant, so that \(B = b_1 N_y + b_2 Q_y\). The remaining part of the annual expenditures (fuel, lubrication, current repairs) is taken to be proportional-

annual output \(E\) kWh and \(Q\) annual heat delivery: \(a'E + cQ\), where \(a'\) is a variable quantity depending on \(Q\), whence

\[ a'E = a_1 E_N + a_2 E_Q = \left(a_1 \frac{E_N}{E} + a_2 \frac{E_Q}{E}\right) E, \]

where \(E_N\) is condensing output, \(E_Q\) is output for heat consumption. Denoting \(E/Q = Z\) and \(E_Q/Q = \Omega\), we obtain

\[ a'E = \left[ a_1 - \frac{\Omega}{Z} (a_1 - a_2) \right] E. \]

Therefore the total annual expenditures for the CHP plant are

\[ A = \left[ a_1 - \frac{\Omega}{Z} (a_1 - a_2) \right] E + cQ + b_1 N_y + b_2 Q_y, \]

and their division between the production of electric power and heat gives

\[ A_E = \left[ a_1 - \frac{\Omega}{Z} (a_1 - a_2) \right] E + b_1 N_y \quad \text{and} \quad A_Q = cQ + b_2 Q_y. \]

Hence the cost of a kWh and of a Mcal is

\[ K_E = a_1 - \frac{\Omega}{Z}(a_1 - a_2) + \frac{b_1 N_y}{E}; \qquad K_Q = c + \frac{b_2 Q_y}{Q}. \]

The shortcomings of B. M. Yakuba’s method are: 1) abandonment of the principle of localizing costs by shops, whereas this principle is dominant in progressive industry, making it possible to approach closely the correct reflection of actual conditions in cost; 2) the difficulty of determining the numerous coefficients \((a, c, a_1, a_2, b_1, b_2)\), which makes it impossible to apply the formulas in practical conditions, especially since the constancy of these coefficients is not maintained even for one and the same type of unit, because their heat characteristics have points of inflection; 3) the questionable and conventional character of the adopted proportionalities: for example, the maintenance of personnel is considered entirely proportional to “no-load operation,” whereas the maintenance of part of the personnel undoubtedly depends on the generation of electric power and the delivery of heat. Likewise, current-repair expenses are caused not only by operation, but also by unit standstills and no-load operation. Fuel and lubricant consumption is also inherent in “no-load operation.” This part of Prof. Yakuba’s proposals is poorly linked with the actual process at a CHP plant. On the basis of what has been said, it may be asserted that B. M. Yakuba’s method can hardly find application in practice.

The analysis made above of a number of proposals for determining the cost of CHP energy products makes it possible to formulate certain basic requirements that should be imposed on a properly constructed method of such determination, namely: 1) the method must reflect the actual process at the CHP plant with sufficient accuracy; 2) it must be constructed according to the principle of cost localization; 3) it must be not difficult in practical application and linked with the principal indicators of the CHP plant, the determination of which is obligatory when there is

...of its rational operation. The rational method seems to be the one that will be based on the basic heat balance of the CHP plant (4)

\[ \frac{860\mathcal{E}+W}{\eta} = \frac{860\mathcal{E}_m+W}{\eta_m} + \frac{860(\mathcal{E}-\mathcal{E}_m)}{\eta_k}. \]

It is immediately evident from this that the consumption of fuel used by the heat consumers over the time interval under consideration, in tons, will be

\[ B_m=\frac{W\cdot 10^{-3}}{\eta_m\cdot Q_p^{p}} \tag{32} \]

Let the total expenditure for the boiler house with the fuel-transport shop over the same interval of time be \(R_{kot}\) rubles, including expenditures on fuel, production expenses, and capital charges (depreciation, repair, insurance). The boiler house produces a single commodity—tons of steam, irrespective of their subsequent purpose; the quantity of these tons is proportional to the number of tons of fuel burned at the specified initial parameters of the steam, feed-water temperature, calorific value of the fuel, and efficiency of the boiler installation. The latter, for modern boilers, is sufficiently stable within considerable limits of variation of their steam output. A certain part of the tons of this steam is then branched off for the needs of heat consumers; moreover, to satisfy them \(B_m\) tons of fuel are required, and therefore, of the total \(R_{kot}\) expenditures, the share of the heat consumers for the boiler-fuel-transport shop will be, in rubles,

\[ R_{kot}^{m}=R_{kot}\cdot B_m/B, \tag{33} \]

where \(B\) is the total fuel consumption at the CHP plant in tons. Further using the principle of localization of costs, we must add to the monetary costs of the boiler house the additional costs \(R_{tf}\) of the district-heating installation: 1) capital charges (depreciation, repair, etc., of boilers, steam converters, district-heating pipelines within the CHP plant, pumping units, the occupied part of buildings, etc.) and 2) its production expenses (personnel, cost of pumping the heat carrier, lubrication, etc.), so that the total expenses falling on heat consumers are \(R_m=R_{kot}^{m}+R_{tf}\).

If during the period of time under consideration \(\dfrac{W}{10^6}\) megacalories were produced, then the cost at the CHP plant of one produced megacalorie is

\[ r_m=\frac{R_m\cdot 10^6}{W}\ \text{rub.} \tag{34} \]

For the electric consumers, however, with total CHP-plant expenditures \(R\), there will remain \(R-R_m=R_e\) rubles, and the cost of a kWh is \(r_e=R_e/\mathcal{E}\) rubles. Thus it turns out that the last method is connected with the heat balance of the CHP plant and its basic efficiencies. This method is based on the principle of localization of costs and, most importantly, seeks to reflect the actual picture of the processes at the CHP plant.^1)

^1) Less convenient would be the determination of the share of fuel falling on electric consumers

\[ B_e=\left(\frac{\mathcal{E}-\mathcal{E}_m}{\eta_k}+\frac{\mathcal{E}_m}{\eta_m}\right)\cdot\frac{860}{Q_p^{n}}, \]

since the calculation of \(\mathcal{E}_m\) in the presence of steam extractions of different pressures for external consumers appears to be a more difficult matter.

Nevertheless, objections are raised against the application of such a method. The cost of a megacalorie is

\[ r_m=\frac{10^6}{W}\left(R^{m}_{kom}+R_{m\phi}\right) =\frac{10^3\cdot R_{kom}}{B\cdot \eta_m Q^p_n} +\frac{10^6\cdot R_{m\phi}}{W} \]

i.e., it proves to be independent of the parameters of the extracted steam. This latter circumstance is advanced by many power engineers as a serious objection to the method described above for determining the cost of energy products.^1) Meanwhile, if one looks closely at the actual process at a heat-and-power plant, it cannot be otherwise here, since for heating purposes what is used mainly is the latent heat of vaporization of the extraction steam, which, as is known, depends little on the pressure of this steam. Utilization of steam of higher pressure only reduces the generation for heat consumption, thereby increasing the condensing generation at the given value of total generation in kWh \(E\), whereby the efficiency \(\eta\) of the heat-and-power plant decreases and, correspondingly, fuel consumption at the heat-and-power plant increases. If, at a low pressure of the extraction steam \((1.2—1.5\ ata)\) and a heat-and-power-plant efficiency \(\eta_1\), the fuel consumption was

\[ B_1=\frac{860\cdot E+W}{\eta_1\cdot Q^p_n}, \]

then, when a consumer with a need for extraction steam of higher pressure \((6—7\ ata)\) is connected, the heat-and-power-plant efficiency will be \(\eta_2<\eta_1\), and the fuel consumption

\[ B_2=\frac{860\cdot E+W}{\eta_2\cdot Q^p_n}>B_1\ kg. \]

The cost of the difference in fuel consumption \(B_2-B_1\) can always be charged, in full agreement with reality, to an increase in the cost of a megacalorie for this consumer with the higher pressure of extraction steam. Thus, without falling into contradiction with the laws of nature, we can understand the struggle against equalization in determining the cost of a megacalorie of extraction steam of different parameters.

Further, in the objections it is often also indicated that the practice of distributing fuel according to formula (33) leads to the result that the principal share of the fuel is charged to thermal energy, greatly increasing its cost, and that this does not contribute to the development of heat-and-power plants. It must be said, first of all, that the development of heat-and-power plants is conditioned by the profitability of the combined production of heat and electric power, of this process as a whole, and in no way can it depend on the partial costs of the energy products. In addition, one senses here a tendency to regard heat converted into electric power as having, as it were, greater value than heat for heating purposes. This tendency, as we have seen, is poorly substantiated. From the standpoint of the national economy, both heat and work satisfy equally important human needs. But the value of one commodity or another is determined not even by this, but by the quantity of socially necessary time required for the production of this commodity—in the present case, heat and work. We have no grounds for considering calories that have left the boiler installation to have different costs, irrespective of their subsequent use. Calories of live

^1) A. N. Rumyantsev wrote, for example, in his brochure Technical Indicators and Costing of the Products of Thermal Power Stations (1935, Energoizdat, p. 9): “Having recognized the principle of different values of heat in all its parameters, we must recognize that high-pressure steam may be cheaper than low-pressure steam. Such an assertion cannot fit into our consciousness.”

steam from the boiler and the calories of reduced steam from the boiler are, of course, equivalent. But after all, an engine whose exhaust steam is utilized for one need or another may also be regarded as a reducer. This exhaust steam, as is most often the case, is directed partly to district-heating needs and partly—for further generation of electric energy in the low-pressure part of the engine; moreover, the proportion of steam consumption for these and other needs changes, and the exhaust steam may go either entirely to the generation of electric energy, or almost entirely to district heating; here there is, as it were, complete interchangeability. In fact, however, there is a difference between these two paths of utilization of exhaust steam from the point of view of the cost of the calories used. Indeed, although the exhaust steam has the same cost for each calorie, regardless of its further use, nevertheless, since the use of each calorie in the low-pressure part of the engine is less perfect (loss in the condenser), for each calorie produced here in the form of electric energy there is a larger quantity of expended calories, which, understandably, will considerably raise the cost of the calorie produced. We shall have the same picture also with respect to the entire condensing part of the heat-and-power station. But this phenomenon is reflected in the considerations set forth above, where

the price of 1 kWh was determined as

\[ \frac{R_{\mathrm{e}}}{\mathcal{E}}=\frac{R-R_{m}}{\mathcal{E}}, \]

i.e., all expenditures were charged to electric energy after deducting the expenditures falling on the heat consumers.

Thus, if we take all these considerations into account, we must arrive at the conclusions that: 1) the independence of the cost of a megacalorie from the parameters of the extraction steam follows from the laws of nature; 2) the calories of heat and of work in an installation with complete utilization of exhaust heat have the same value, for in the boiler installation the same quantity of socially necessary time is expended for the production of each of them.

In conclusion it must be said that, in cost, we are obliged to reflect the real process at a heat-and-power station, whatever the results of this may be: one must not, in the name of subsidiary considerations, distort reality in order to fit it to considerations invented out of one’s head or extraneous to the process. This would be an idealistic approach with all the consequences that may follow from it. F. Engels already pointed out that the forms of being thought can never derive from itself, but only from the external world: nature does not conform to principles; principles are true only insofar as they correspond to nature.^1) From this point of view we should not be troubled even if the specific fuel consumption and the cost per megacalorie should prove higher in the case of extraction steam than for steam from separate boiler houses.^2) If the cost should also prove higher, it must never be confused with the selling price per megacalorie, which may be set at any level depending on the conditions of general policy on the energy front at the given stage of development of the national economy, provided only that the general benefit from the use of heat-and-power stations is preserved.

^1) F. Engels, Anti-Dühring, Partizdat, 1933, p. 24.

^2) ORGRES of the NKES, in a letter addressed to the author dated 25/IV–41, wrote: “The attribution of part of the electrical and mechanical losses in the turbine installation to the heat supplied to heat consumers is connected with the fact that the specific fuel consumption per megacalorie when heat is extracted from the turbine will be higher than when steam is supplied from a boiler house.”

References

1. Hazen—A new method for assessing the perfection of steam-power installations and the distribution of heat-flow streams at combined heat-and-power plants. Heat and Power, No. 2, 1936.

2. Lagovskii—Heat-Power Installations, p. 201, 1932.

3. Standardization of Steam Turbines and Thermal Schemes of Large Electric Power Stations, part 1, 1933, pp. 157–160, 1933.

4. Butakov—Determination of the diameter of the passage of steam pipelines according to the maximum of their efficiency for condensing electric power stations. Bulletin of Engineers and Technicians, 1938, No. 11.

5. Butakov—Steam Pipelines of Electric Power Stations: Their Design, Schemes, Calculation, pp. 130–146, 1932.

6. Dollezhal—Fundamentals of the Design of Steam-Power Installations, p. 19, 1943.

7. The Heat-Power Economy of Railway Transport, 1940, vol. 1, part 1.
   Luknitskii—Heat-Power Installations, p. 68.

8. Fuks—A new method for determining the degree of perfection of a heat-power installation. Proceedings of the Energy Conference. Tomsk, pp. 7–11, 14–17, 1941.

9. Gorshkov—Methods of standardization, accounting, and analysis of energy balances of thermal power stations by the turbine hall. Moscow, pp. 78–87, 1939.

10. Herzen—Selected Philosophical Works, vol. 1, p. 98, 1946.

11. Yakub—Steam Turbines, p. 42, 1935.

12. Tumasov—Techno-economic indicators of combined heat-and-power plants. Electric Power Stations, 1937, No. 5.

13. See 2.

14. Yanovskii—On a new method for assessing the perfection of a steam-power installation. Soviet Boiler Construction, 1937, No. 5.

15. Yurinskii—Thermodynamic foundations for comparing heat-power installations. Proceedings of the Tomsk Polytechnic Institute, 1944, vol. 63.

16. Ryzhkin—On the question of determining the cost of a ton of steam and kWh at installations with combined production of thermal and electrical energy. Electric Power Stations, 1930, No. 6, p. 365.

17. Rumyantsev—Technical Indicators and Costing of the Output of Thermal Electric Power Stations, p. 22, 1933.

18. See 7, p. 112.

19. Andryushchenko—On the question of the cost of electrical and thermal energy when they are produced jointly at combined heat-and-power plants. Soviet Boiler Construction, 1940, No. 4.

20. Yakub—Thermal Electric Power Stations, 1938, vol. 1, pp. 40–42.

Observed Misprints

Page	Line	Printed	Should be
5	3 from bottom	\(AEIGA\)	\(ABCDEA\)
11	11 from bottom	\(\displaystyle \frac{\eta_m}{\eta_k\ \text{or}\ \eta_m}\)	\(\displaystyle \frac{\eta_m}{\eta_k\ \text{or}\ \eta_n}\)
16	15 from bottom	\(\displaystyle \frac{1}{\eta_m}\cdot \frac{860\varphi}{(W_k' W)q}\)	\(\displaystyle \frac{1}{\eta_m}\cdot \frac{860\varphi}{\left(1+\frac{W_k}{W}\right)q}\)
19	10 from bottom	\(\displaystyle \frac{C_2D_{otb}}{860}\)	\(\displaystyle -\frac{C_2D_{otb}}{860}\)
24	1 from top	\(\mathcal{E}'_m\)	\(\mathcal{E}''_m\)
24	4 from top	\(\displaystyle \left(1-\frac{\mathcal{E}'_m}{\mathcal{E}}\right)\)	\(\displaystyle \left(1-\frac{\mathcal{E}''_m}{\mathcal{E}}\right)\)
24	11 from bottom	\(\displaystyle \frac{860(i'_n-t_{kond})}{\eta''_e(i'_1-i''_n)}\)	\(\displaystyle \frac{860(i''_n-t_{kond})}{\eta''_e(i'_1-i''_n)}\)
42	6 from top	\(Q_p\)	\(Q_p^n\)
48	5 from bottom	linii	liniy
150	16 from top	\({}'G_0^{ist}\)	\(x'_2G_0^{ist}\)
150	20 from top	\(s\)	\(s_1\)
151	3 from top	\(x_{10}'2\)	\(x_{10}^{\prime 2}\)
151	6 from top	\(x_{10}^{12}\)	\(x_{10}^{\prime 2}\)

1. In Nos. 11--12 of the journal For Fuel Economy for 1945 there appeared a note testifying that the existing system of bonuses for fuel economy at the NKES does not take into account the output of hydroelectric stations, which leads to obvious absurdities in practice: a group of personnel connected with the operation of a hydroelectric station is deprived of bonuses when the load of the hydroelectric station is increased, because the operating conditions of the thermal stations deteriorate, although the increase in the specific fuel consumption at the latter is many times compensated by the fuel economy from reducing the output of thermal stations. This absurdity could not occur if, in evaluating the power system in terms of fuel consumption, the above considerations were used, giving a broader mediation of reality. ↩