Thermal Engineering Basis for Comparing Heat-Power Installations

V. T. Yurinsky

Associate Professor, Candidate of Technical Sciences

Introduction

From the moment when James Watt produced the first steam engine built according to his patent, the heat engine entered upon a path of technical development determined not only by reliability of operation, but also by economy.

The striving for economical ways of using the heat of fuel led to the idea of carrying out a combined energy process, i.e., to district heating. In this case the thermal power installation is replaced by a heat-power installation. A heat-power installation has as its task the simultaneous supply of an energy district with heat and power. Since this task can be solved in several variants, it is of great interest to develop a universal method for comparing heat-power installations, one that makes it possible to compare with one another the various variants of energy supply for a given district from the standpoint of their thermotechnical advantage. The present work is devoted to consideration of this method.

In particular, the work has as its aim the determination of the conditions under which two installations of different types become equivalent. Thus, here, in a general form, there is solved and examined in detail the problem which Acad. M. V. Kirpichev formulated in 1924 in the work “On the Mixed Use of Heat in Steam Installations” [1].

The second task of the present work is to find a general expression for the efficiency of a heat-power installation in a form that would be universal, i.e., would make it possible to find the efficiency for any heat-power installation or thermal-energy system.

The Energy-Supply District. Choice of Conditions for Comparing Installations of Different Types

By an energy-supply district we shall understand the aggregate of electric (power) and thermal consumers claiming centralized energy supply, i.e., supply with heat and power. It is assumed here that heat must be supplied in the form of steam or hot water.

An energy-supply district may be both very broad and very narrow. For example, the matter may concern the energy supply of the Moscow Region from a single energy system including a number of hydroelectric stations and a number of thermal stations.

At the same time, nothing prevents us, under other conditions, from understanding by a power-supply district an individual plant, an individual city, or even an individual building or group of buildings.

In the first case the discussion will concern power supply from a unified power system; in the second, from a single power center. We may go still further and speak of supplying a district with power from one isolated machine.

It is clear that, in the case of operation within a large power system comprising a considerable number of machines of various types, each of them can be placed under the most favorable conditions. Conversely, supplying a district with heat and power from an isolated machine represents the operating regime most unfavorable for the machine. Here the characteristic features of the machine’s behavior stand out most sharply.

Every power district, comprising a number of heat and electric (power) consumers, entails the existence of quite definite heat and electric loads.

In the case of supply from a power system, these loads must be covered by the units forming part of the system. Depending on the relation between the heat and electric loads at the separate geographically distributed points of the district, and depending on the availability of local fuel bases, hydropower resources, etc., the individual machine units of the system must be located accordingly.

When heat and power are supplied from an isolated installation, the radius of heat supply is small, and in practice the question reduces to the heat supply of some single point on the geographic map of the country. Electric power, however, can be transmitted from an isolated station over considerable distances. Consequently, the discussion may, both theoretically and practically, concern the power supply of an entire geographic district.

The essence of the matter, of course, changes according to what the power district represents from the geographic point of view. For the engineer designing the station itself or the system, the power district represents the aggregate of heat and electric loads that make it possible to determine the capacity of the system only at the very initial stage of design. After this, taking into account the considerations stated above, it is possible to outline the capacity and structure of the individual centers of the system.

Of extraordinary importance is the presence in the district of two types of load: electric and heat.

These types of load differ from one another not only in the possible radius of transmission over distance. By their very nature these loads are usually not interconnected.

A whole series of factors (meteorological conditions, domestic habits, the character of the load diagrams of industrial consumers, etc.) will lead to the emergence of a disproportion between the electric and heat loads, i.e., to such a state of affairs that, in the combined process of power supply, either the heat load is insufficient for the generation, on heat consumption, of all the electric power required by the district, or, conversely, the heat load is so large that it will be necessary to resort to supplying heat to consumers by using reduced live steam that has not first passed through a steam engine.

Let us note that even where a developed power system exists, the emergence of disproportion is not excluded, since heat supply is territorially tied to certain definite centers of the system.

In a power system, an individual center cannot count on support and replacement from other centers so long as the question concerns heat supply. The situation is different with respect to the generation of elec-

…electric energy. The existing electrical interconnections make it possible to redistribute the electrical load among individual centers according to the principle of an economically optimal solution of the problem.

For an individual power center not included in the system, the issue may be the redistribution of heat and electrical loads among the individual units of that center.

Finally, when we are dealing with a separately operating unit, that unit must be counted only on its own, both from the standpoint of covering the electrical load and from the standpoint of covering the heat load.

In this case there is not even occasion to raise the question of how economical it will be to accept the heat and electrical loads existing at each given moment. They cannot be left unmet; otherwise the heat-power installation will not perform its principal task—satisfying the needs of the consumers connected to it.

From what has been said it follows with sufficient clarity that the existence of a disproportion between the heat and electrical loads will have the most serious effect on an isolated unit. Here it is easiest to incur an overconsumption of fuel as a result of an unfortunate combination of the two kinds of load.

Consequently, in order to judge correctly the behavior of one or another type of machine in performing its task of supplying energy to a district, it is necessary to single out an isolated unit. The same unit, when incorporated as an element into a power system consisting of units of different types, or even of the same type, will be placed under more favorable conditions. Of course, in practice it is not so simple to resolve the question of distributing the load among the individual units of a system. In each specific case the required load-distribution schedule will have to be found either empirically or by attempting to find a mathematical solution of the problem.

It is not difficult to see that the case in which a unit operates separately from the system is also the most general one. Indeed, a single unit may be considered both as a separate unit and as an element of a system.

Therefore, in what follows we shall compare isolated machines of different types, speaking of different variants of energy supply for a district.

Types of Installations and the Method for Comparing Them

At present, three basic types of turbines are used in heat-power installations: 1) condensing, 2) back-pressure, and 3) extraction-condensing turbines.

In accordance with this, we shall compare installations in which electrical consumption is covered by the operation of one of these three types of engines. We shall compare the variants of separate energy supply, energy supply from a back-pressure turbine, and energy supply from an extraction turbine. The method of comparison will consist in the following: taking, for one and the same energy district, two types of installations, we shall determine the conditions under which both installations become equivalent from the standpoint of the heat expenditures required for their operation.

In doing so, we remove from consideration the factors that do not depend on the type of machine installation. Such factors are the direct heat losses in the boiler house, which occur during fuel combustion and depend not on the purpose for which the steam delivered by the boiler house is used, but on the grade of fuel, the method of its combustion, the type of boiler, and other such factors.

The losses in station steam lines and the losses in the external heat network are excluded from consideration on the grounds that they do not depend on…

type of machine units. Thus, the quantity of heat expended on the production of steam in the boilers, without taking into account losses in the boiler plant and steam pipelines, is taken as 100%. As will be shown in the last section, the transition to an absolute index of the economy of a heat-power installation makes it possible, without difficulty, to take these losses into account as well.

When comparing installations of two types, we always take as an obligatory premise the consideration that either of the two installations being compared must fully cover the district’s heat and electric loads.

Comparison of Separate Power Supply with Power Supply from a Back-Pressure Turbine

The variants of power supply considered in this chapter are of greatest importance for small installations (power stations of small enterprises, small municipal stations, compressors, turbo-blower stations, etc.). In our opinion, when selecting the type of engines for installations of this kind, in those cases where the power-supply district necessitates covering both heat and power (electric) loads, the variant of installing back-pressure turbines must necessarily be considered; moreover, this variant is among the competitive ones for the following reasons.

1. The capital expenditures for it are minimal.
2. As will be shown below, in a number of cases this variant proves advantageous from the thermotechnical point of view as well.

Before proceeding further, we wish to preface the exposition with certain thoughts concerning the operating conditions of back-pressure turbines. The aim we pursue here is to establish a more objective approach to the back-pressure turbine and to reveal the possibilities of this class of machines. In the literature known to us, the possibilities of back-pressure machines are underestimated, and on the basis of this underestimation an often incorrect judgment is made concerning the possibilities of using back-pressure turbines to cover the heat and power loads of a district.

The essence of this underestimation of the question of the economy of operation of back-pressure machines lies in the fact that the starting point is the postulate that operation of a back-pressure turbine is advantageous only according to the heat schedule. It is assumed that operation of a back-pressure turbine according to a free electric schedule is always disadvantageous. The latter is accepted a priori as something entirely indisputable, requiring no proof.

This proposition is advanced most categorically by our Soviet authors. Thus, for example, in the recently published book by A. V. Shcheglyaev [2] we find the diagram for connecting a turbine with back pressure, shown in Fig. 1, which provides for the obligatory parallel operation of the back-pressure generator and condensing turbogenerators on a common electric network. It follows from the text that the only operating condition of a back-pressure turbine can be the condition of covering the electric load according to the heat schedule. The remaining electric load must be covered by condensing machines operating in parallel. Operation of the turbine according to the electric schedule occurs only during those hours when the electric load does not meet the heat load, i.e., when the steam consumption for heat demand exceeds the steam consumption by the turbine necessary to cover the electric load. Obviously, at such times the condensing machines must be unloaded to no-load operation or stopped. The shortage of steam for heat demand is covered by supplying reduced live steam to the network.

The same a priori premise is found in the works of B. M. Yakub [3, 4]. Yakub goes even further than A. V. Shcheglyaev, emphasizing that a back-pressure turbine must always (and consequently also at those moments when Shcheglyaev permits operation according to the electrical schedule) operate according to the heat-load schedule, supplying only such an electrical load as is dictated at the given moment by heat consumption.

Fig. 1

at the given moment. A situation in which the electrical load of the grid is less than the electrical load supplied by heat consumption is excluded from consideration altogether.

Such a conception of the operation of a back-pressure turbine seems to us insufficiently complete. The mode of operation according to the heat-load schedule, indicated by Shcheglyaev and Yakub, is of course possible. In certain cases, for example as applied to a system consisting of a number of stations, with a large electrical load in the district, this will evidently be the most advantageous mode. But the more general case will be the case in which there is a disproportion between the heat load and the power load on the turbine.

First, a back-pressure turbine may be the only prime mover supplying all the heat and all the electrical consumption; moreover, especially as applied to small factory-type stations, this, as will be shown below, in many cases gives the most correct solution of the energy-supply problem from every point of view.

Second, the premise that the turbine is coupled to an electric generator is not obligatory. It may be used as a drive for a blower, compressor, pump, etc. In this case, it goes without saying, the covering of the power load will be presented to the back-pressure turbine as an independent task.

Taking into account what has been said above, a number of authors approach the back-pressure turbine differently, assuming that the specific conditions of the energy (heat and electrical) load of the district may place the back-pressure turbine under the necessity of:

1) operating according to the heat-load schedule,
2) operating according to the electrical-load schedule.

Indications that a back-pressure turbine may be placed under the conditions of one of the above-mentioned modes are found, for example, in—

for example, in a number of sources [5, 6, 15]. Fig. 1 and Fig. 2 show what the difference consists in between operation according to a heat schedule and according to an electric schedule.

It is curious, however, to note that German authors also underestimate the possibilities of the back-pressure turbine, assuming that it can operate economically only in the case where the steam consumption for heat demand exceeds the steam flow through the turbine caused by the electric (power) load. (The highest economy is achieved with strict matching of the heat and electric loads.) If the opposite relation between the heat and power loads is observed, then the operation of the turbine already appears unfavorable.

Fig. 2

K. Tsiteman expresses this idea in a cautious form.

Ernst states it more categorically in the following words: “Single operation of a back-pressure turbine cannot proceed favorably in the economic sense if the steam consumption for the production of energy exceeds the steam consumption for heat demand. In this case the best means for equalizing the electric load is parallel operation with an outside network of large capacity, capable both of receiving and of delivering power.” Thus Ernst, too, arrives at the same idea of the necessity of operating a back-pressure turbine according to a free heat schedule and a forced electric schedule, i.e., he takes the same position as Yakub and Shcheglyaev.

To what does the above-noted view of the disadvantage of operating a back-pressure turbine according to a free power (electric) load schedule lead? Obviously, if we agree with it, we must a priori accept the following formula:

“If all the steam that has passed through the turbine cannot be wholly absorbed by the heat consumer, then the use of a back-pressure turbine is inadvisable.”

An illustration of such a view is, for example, the work of Eng. S. V. Balyan [7].

Eng. Balyan arrives in a number of cases at incorrect conclusions precisely because, as his criterion for the advisability of using a back-pressure turbine, he takes the mismatch visible to the naked eye—

the ability of the heat consumer to absorb all the steam that has passed through the turbine.

Indeed, if we imagine a back-pressure plant discharging surplus steam at a sufficiently high pressure into the atmosphere, we arrive, purely psychologically, at the conclusion that such a plant is very imperfect; the heat losses in it are large and the economic effect is low. On the basis of this purely emotional perception of things we do not consider it necessary to penetrate into the essence of the matter and refrain from a deep analysis.

We may assert that, even in the presence of a disproportion between the heat and electric (power) loads—a disproportion lying within certain limits—a back-pressure turbine may advantageously prove to be a competitor to the separate power-supply variant, contrary to ordinary “common sense,” which cannot accept the fact that discharging relatively large quantities of steam at a sufficiently high pressure into the atmosphere may be economically advantageous.

In using the term “discharge into the atmosphere,” we are not inclined to understand it literally. It will be rational to apply schemes of back-pressure plants that provide for the use of the condensate from the discharged surplus steam. The preservation of the condensate is, of course, very important, if only from the standpoint of the water-treatment problem.

This latter circumstance will be taken into account by us when deriving the fundamental relationships, for the use of the condensate is practicable and expedient.

Thus, we wish to show that, from the standpoint of fuel economy, the use of a back-pressure machine may in a number of cases be advantageous. The question of the thermal-engineering expediency of using this type of machine is decided not by the fact of discharging surplus steam into the atmosphere, but by the ratio between the quantity of steam used for heat consumption and the quantity of steam that has passed through the turbine.

Let us consider two types of plants intended to satisfy the heat and electric loads of a district. The two plants being compared are shown schematically in Fig. 3 and Fig. 4. Fig. 4 shows a back-pressure plant intended for the simultaneous satisfaction of the needs of heat and electric consumers. Fig. 3 gives the scheme of a plant that consists of a boiler and a condensing turbine. The condensing turbine provides the necessary generation of electric (power) energy; the heat load of the district is covered by the boiler’s delivery of fresh steam, passed through a reducing valve.

Let us plot on the \(i—s\) diagram (Fig. 5) representations of the following processes: 1) the process in the condensing turbine, 2) the process in the back-pressure turbine, 3) the conditional process of throttling steam in the reducing valve. In depicting these processes it is assumed that:

1) the relative internal efficiency of the turbine with back pressure is higher than the efficiency of the high-pressure cylinder of the condensing turbine of the same name, owing to the greater steam flow of the back-pressure turbine, which leads to more favorable operating conditions of the flow part in the high-pressure region (from \(p_0\) to \(p_{\text{п}}\));

2) the initial parameters are taken to be identical for both types of turbines and for the throttling process.

Let us introduce the following notation:

\(D_{\text{к}}\) — hourly steam consumption of the condensing turbine, kg/hour (excluding auxiliary requirements);

\(D_{\text{п}}\) — hourly steam consumption of the back-pressure turbine, kg/hour (excluding auxiliary requirements);

$D_{\mathrm{p}}$ — hourly steam consumption by the back-pressure turbine, including the station’s auxiliary requirements;

$\eta_{0ik}$ — relative internal efficiency of the condensing turbine;

$\eta_{0ip}$ — relative internal efficiency of the back-pressure turbine.

In accordance with what has been said above, we assume that the condensing or back-pressure turbine must cover (each separately) the electric (power) load of the district. This condition is absolutely necessary, since we are comparing two machines intended for the power supply of one and the same consumption district (power district).

Fig. 3

Fig. 4

For a condensing engine the power equation is written as follows:

\[ N_e^{k}=\frac{D_k(i_0-i_k)\eta_{mk}\eta_{gk}}{860}. \tag{1} \]

For a back-pressure turbine we have, analogously:

\[ N_e^{p}=\frac{D_p(i_0-i_p)\eta_{mp}\eta_{gp}}{860}. \tag{2} \]

In both variants the electrical power of both turbines, required to cover the load of the district, must be the same, which is expressed by the notation

\[ N_e^{k}=N_e^{p} \tag{3} \]

Substituting in equation (3) the values of \(N_e^{k}\), \(N_e^{p}\) from equations (1) and (2), we may write:

\[ D_k(i_0-i_k)\eta_{mk}\eta_{gk} = D_p(i_0-i_p)\eta_{mp}\eta_{gp} \tag{4} \]

From equation (4) we determine \(D_k\) in terms of \(D_p\)

\[ D_k = D_p \frac{(i_0-i_p)\eta_{gp}\eta_{mp}} {(i_0-i_k)\eta_{gk}\eta_{mk}}. \tag{5} \]

Fig. 5

Let us assume that the condensate of the steam supplied for heat consumption is returned in full with heat content \(i_2\) kcal/kg.

Let the condensate entering the boiler feed from the condensing turbine have heat content \(i_2'\) kcal/kg. Let us determine the amount of heat delivered by the heat network of the power district. Obviously, both in the case of a separate installation and in the case of a back-pressure turbine, the same amount of heat \(W\) kcal/hr must be released to the network. In the case of a separate installation, the boilers must release to the network \(D_n\) kg/hr of steam, reduced to pressure \(p_p\) (Fig. 5), but retaining the initial heat content \(i_0\). Assuming that the condensate return occurs under the same conditions as the condensate return with the back-pressure turbine (the same pressure in the network), we determine the amount of heat delivered to the network,

\[ W=D_n(i_0-i_2)\ \text{kcal/hr} \tag{6} \]

For a back-pressure turbine it is necessary to take into account the following circumstance. All the steam passing through the turbine is divided into two streams

\[ D_p'=D_k+D_n', \tag{7} \]

where \(D_p'>D_p\); \(D_n'>D_n\); \(D_k'>D_k\). This fundamentally takes into account the steam consumption for the plant’s own needs.

$D_{\mathrm{n}}'$—the amount of steam used to cover the heat demand;

$D_{\mathrm{a}}'$—the amount of surplus steam “discharged” into the atmosphere.

We assume, therefore, that the electric load of the district is not linked with the heat load. The heat-consumption base is insufficient for the entire amount of steam that has passed through the turbine—the amount determined by the electric load—to be used by the heat network. The surplus steam can find only one application: its use to preheat the make-up feedwater to the maximum possible temperature, determined by the pressure of the steam leaving the turbine. This possibility, as one that suggests itself, will be taken into account in the subsequent constructions.

In accordance with what has been said, the amount of heat delivered by the back-pressure turbine to the consumption network will be

\[ W = D_{\mathrm{n}}'(i_{\mathrm{n}} - i_2). \tag{8} \]

The amount of heat lost is

\[ Q_{\mathrm{n}}' = D_{\mathrm{a}}'(i_{\mathrm{n}} - i_2). \tag{9} \]

Equating the right-hand sides of equations (6) and (8), we find:

\[ D_{\mathrm{n}}(i_0 - i_2) = D_{\mathrm{n}}'(i_{\mathrm{n}} - i_2), \tag{10} \]

whence

\[ D_{\mathrm{n}} = D_{\mathrm{n}}'\frac{i_{\mathrm{n}} - i_2}{i_0 - i_2}. \tag{11} \]

For an objective comparison of the two variants of energy supply, it is also necessary to take into account, in concrete form and not only in principle, the expenditure of energy for auxiliary needs. With respect to auxiliary consumption, each of the installations under consideration has its own specificity.

For one and the same electric load of the district, the consumption of electric power, and consequently also the consumption of heat (steam) for its generation, will be different for the two installations being compared.

First, owing to the fact that

\[ D_{\mathrm{n}} + D_{\mathrm{k}} \ne D_{\mathrm{p}}, \]

even at the same initial pressure the expenditures of electric power on the auxiliary motors of the mechanisms providing steam production (smoke exhausters, fans, feed pumps, etc.) will be different.

Second, the expenditures of electric power for the auxiliary needs of the turbine hall will also be unequal. In the case of a condensing turbine it is necessary, at the expense of electric-power expenditure, to ensure the operation of the condensing group of pumps. In the case of a back-pressure turbine discharging surplus steam into the atmosphere (in the literal sense of this), there is, of course, no item of auxiliary expenditure connected with the condensation of steam that has left the turbine and has not been delivered to heat consumers. In more advanced schemes of back-pressure turbines, providing for the preservation of condensate in an amount corresponding to the passage of steam through the turbine, an additional expenditure of energy will be required to carry out the expedient task posed. This expenditure will be a specific feature of this type of installation. Of course, the expenditures of electric power for auxiliary needs with back-pressure and condensing turbines will also differ in the turbine-hall line, and here the advantage will be on the side of the former.

Thus, with respect to two items of auxiliary-service expenditure (the boiler house and the turbine hall), the compared variants of district power supply will be non-equivalent.

The third item of auxiliary-service expenditure is the expenditure of energy on pumps that provide heat supply to the district (network pumps, booster pumps). A number of authors assign this item to network expenditures. Others divide the expenditure of electric power associated with the operation of the network into two parts, treating the energy expenditure for pumps installed in the station building as station expenditure, and the energy expenditure for pumps located at a distance over the territory as network expenditure.

Finally, it is possible to regard the expenditure of electric power on the heat network as an external electric load of the district and, on this basis, to exclude this group of expenditures altogether from consideration when comparing power-supply variants.

It seems to us most correct to assign all the electric-power expenditure associated with operation of the network to the station auxiliary services, since, depending on the type of the station installation itself, this expenditure may be different.

Assigning the electric expenditure for the heat network to the station auxiliary services is correct already because, as the installation being compared, there may be an installation of a turbine with deteriorated vacuum, the motors of whose pumping group we shall treat, of course, as motors of the station auxiliary services.

We shall assume further that any of the turbines being compared must itself cover the auxiliary-service load. This means that the power developed by the turbine at each given moment will be different for the two variants.

For the condensing turbine the required power will be:

\[ N_{\mathrm{e}}^{\mathrm{k}} + N_{\mathrm{sn}}^{\mathrm{k}}, \]

for the back-pressure turbine:

\[ N_{\mathrm{e}}^{\mathrm{p}} + N_{\mathrm{sn}}^{\mathrm{p}}, \]

where \(N_{\mathrm{e}}^{\mathrm{k}} = N_{\mathrm{e}}^{\mathrm{p}}\) is the electric load of the district,

\(N_{\mathrm{sn}}^{\mathrm{k}}, N_{\mathrm{sn}}^{\mathrm{p}}\) is the auxiliary-service load of the installations.

\[ N_{\mathrm{sn}}^{\mathrm{k}} \ne N_{\mathrm{sn}}^{\mathrm{p}}. \]

The power required to cover the auxiliary services in each of the variants is generated by means of the same process as that used to cover the district electric load. To ensure the generation of electric power for the auxiliary services, an additional quantity of steam must be passed through the turbine. We shall denote this additional steam consumption as \(D_{\mathrm{k}}^{\mathrm{sn}}\) for the separate power-supply variant, and as \(D_{\mathrm{p}}^{\mathrm{sn}}\) for the back-pressure-turbine variant. It is clear that, for the back-pressure turbine, this steam flow too will be divided into two parts, \(D_{\mathrm{k}}^{\mathrm{sn}}\) and \(D_{\mathrm{l}}^{\mathrm{sn}}\). The quantity of steam used by the heat network is

\[ D_{\mathrm{k}}' = D_{\mathrm{p}} + D_{\mathrm{p}}^{\mathrm{sn}} - D_{\mathrm{l}} - D_{\mathrm{l}}^{\mathrm{sn}}. \]

Let us determine the heat expenditure for both variants.

A. Separate power supply.

The quantity of steam delivered by the boilers is

\[ D_1 = D_{\mathrm{k}} + D_{\mathrm{k}}^{\mathrm{sn}} + D_{\mathrm{n}}. \tag{12} \]

The quantity of heat expended on the production of steam is

\[ Q_1 = D_{\mathrm{k}}(i_0 - i_2') + D_{\mathrm{k}}^{\mathrm{sn}}(i_0 - i_2') + D_{\mathrm{n}}(i_0 - i_2). \tag{13} \]

B. Back-pressure turbine variant.
The quantity of steam delivered by the boilers, and, correspondingly, the quantity of heat expended on the production of steam, is

\[ D_2 = D_{\mathrm{p}} + D_{\mathrm{p}}^{\mathrm{sn}}, \tag{12} \]

\[ Q_2 = D_{\mathrm{p}}(i_0 - i_2) + D_{\mathrm{p}}^{\mathrm{sn}}(i_0 - i_2). \tag{13} \]

Let us assume that the second variant gives a heat saving in comparison with the first.

The magnitude of the saving is obtained by subtracting equation (13) term by term from equation (13)

\[ E = Q_1 - Q_2. \tag{14} \]

\[ E = D_{\mathrm{k}}(i_0 - i'_2) + D_{\mathrm{k}}^{\mathrm{sn}}(i_0 - i'_2) + D_{\mathrm{n}}(i_0 - i_2) - D_{\mathrm{p}}(i_0 - i_2) - D_{\mathrm{p}}^{\mathrm{sn}}(i_0 - i_2). \tag{15} \]

Equation (15) shows that the saving may be either positive or negative, depending on the ratio of the district’s electrical and heat loads, and also on the consumption of electric power for station auxiliaries in each of the variants.

The two installations—separate power supply and back-pressure—become equivalent when the heat saving is reduced to zero, i.e., under the condition

\[ E = Q_1 - Q_2 = 0. \tag{16} \]

However, condition (16) may be replaced by another, equivalent one. The installations become equivalent only when, in covering identical heat and electrical loads, we have heat losses of equal magnitude in both cases.

Let us write this condition as

\[ \Sigma Q'_{\mathrm{loss}} = \Sigma Q''_{\mathrm{loss}} \tag{17} \]

\(\Sigma Q'_{\mathrm{loss}}\) — sum of losses in the separate variant,
\(\Sigma Q''_{\mathrm{loss}}\) — sum of losses in the back-pressure variant.

For the separate power-supply variant,

\[ \Sigma Q'_{\mathrm{loss}} = D_{\mathrm{k}}(i_{\mathrm{k}} - i'_2) + D_{\mathrm{k}}^{\mathrm{sn}}(i_0 - i'_2) + D_{\mathrm{k}}(i_0 - i_{\mathrm{k}})(1 - \eta_{\mathrm{m k}}\eta_{\mathrm{g k}}). \tag{18} \]

For the back-pressure variant,

\[ \Sigma Q''_{\mathrm{loss}} = D_{\mathrm{p}}(i_{\mathrm{p}} - i_2) + D_{\mathrm{p}}^{\mathrm{sn}}(i_0 - i_{\mathrm{p}}) + D_{\mathrm{p}}^{\mathrm{sn}}(i_{\mathrm{p}} - i_2) + D_{\mathrm{p}}(i_0 - i_{\mathrm{p}})(1 - \eta_{\mathrm{m p}}\eta_{\mathrm{g p}}). \tag{18'} \]

These expressions reflect the circumstance that, in essence, the heat expenditures for covering the electrical auxiliary requirements of the installation constitute an additional loss of the installation.

Since the electric energy for satisfying the station’s own requirements is produced in the same turbogenerator, i.e., with the same heat drop utilized and at the same values of \(\eta_{\mathrm{m}}\) and \(\eta_{\mathrm{g}}\), we may write

\[ N_{\mathrm{k}}^{\mathrm{sn}} = r_1 N_{\mathrm{e}}^{\mathrm{k}} \]

\[ N_{\mathrm{p}}^{\mathrm{sn}} = r_2 N_{\mathrm{p}}, \]

or

\[ \left. \begin{aligned} D_{\mathrm{k}}^{\mathrm{sn}} &= r_1 D_{\mathrm{k}},\\ D_{\mathrm{p}}^{\mathrm{sn}} &= r_2 D_{\mathrm{p}}, \end{aligned} \right\} \]

similarly

\[ \left. D_{\mathrm{l}}^{\mathrm{sn}} = r_2 D_{\mathrm{l}}, \right\} \tag{19} \]

where \(r_1, r_2\) are relative figures indicating the consumption of electric power or steam for auxiliaries.

Using the notations (18), (18′), (19), we expand equation (17)

\[ D_k(i_k-i'_2)+r_1D_k(i_0-i'_2)+D_k(i_0-i_k)(1-\eta_{mk}\eta_{gk}) = D_l(i_p-i_2)(1+r_2)+ \]
\[ {}+r_2D_p(i_0-i_p)+D_p(i_0-i_p)(1-\eta_{mp}\eta_{gp}). \]

Replacing in this equation \(D_k\) by \(D_p\) from (5), after a number of transformations we obtain

\[ D_p\left[ \frac{i_0-i_p}{i_0-i_k}\cdot \frac{\eta_{mp}\eta_{gp}}{\eta_{mk}\eta_{gk}} (i_0-i'_2)(1+r_1) - (i_0-i_p)(1+r_2) \right] = \]
\[ = D_l(i_p-i_2)(1+r_2) \tag{20} \]

Introduce the notation

\[ \varepsilon=\frac{D_p}{D_l} \tag{21} \]

and determine \(\dfrac{D_p}{D_l}\) from equation (20)

\[ \varepsilon=\frac{D_p}{D_l} = \frac{(i_p-i_2)(1+r_2)} { \dfrac{i_0-i_p}{i_0-i_k}\cdot \dfrac{\eta_{mp}\eta_{gp}}{\eta_{mk}\eta_{gk}} (i_0-i'_2)(1+r_1) - (i_0-i_p)(1+r_2) }. \tag{22} \]

With a different grouping of terms, formula (22) may be written in the form

\[ \varepsilon= \frac{(i_p-i_2)(1+r_2)} { \dfrac{i_0-i_p}{i_0-i_k} \left[ \dfrac{\eta_{mp}\eta_{gp}}{\eta_{mk}\eta_{gk}} (i_0-i'_2)(1+r_1) - (i_0-i_k) \right] - r_2(i_0-i_p) }. \tag{22′} \]

In the particular case when

\[ \eta_{mp}=\eta_{mk} \]

\[ \eta_{gp}=\eta_{gk} \]

expression (22′) for \(\varepsilon\) is simplified and takes the following form:

\[ \varepsilon= \frac{(i_p-i_2)(1+r_2)} { \dfrac{i_0-i_p}{i_0-i_k} \left[ (i_k-i'_2)+r_1(i_0-i'_2) \right] - r_2(i_0-i_p) }. \tag{22″} \]

In the latter form, the expression for \(\varepsilon\) was also used in solving numerical examples 1, 2, 3, 4.

Equations (22), (22′) show what, under the specified conditions of the process in the condensing and back-pressure turbines, the ratio \(\varepsilon\) must be; this ratio is a criterion for the heat and electric loads of the district.

\(D_p\) is determined by the electric load at the given moment.

\(D_l\) is determined by the heat load at the given moment.

The magnitude \(\varepsilon\), as we see, depends on the degree of technical perfection of the turbines being compared. It also depends on the expenditure of energy for auxiliary needs in each of the installations being compared, as well as on the initial and final parameters of the turbines. Consequently, \(\varepsilon\) also depends on the character of the heat load, since the final parameters of the back-pressure turbine are determined by the requirements of the heat consumers connected to it.

Through the back-pressure turbine there is passed the quantity of steam (allowing for auxiliary requirements)

\[ D'_n = D'_n + D'_l . \]

Since

\[ D'_n = (1 + r_2)D_n, \]

\[ D'_l = (1 + r_2)D_l, \]

then also

\[ \varepsilon = \frac{D_n}{D_l} = \frac{D'_n}{D'_l}. \]

The quantity of steam used from the total steam flow by the back-pressure turbine is

\[ D'_i = D'_n - D'_l = D'_n - \frac{D'_n}{\varepsilon} = D'_n\left(1 - \frac{1}{\varepsilon}\right), \]

\[ D'_i = \frac{\varepsilon - 1}{\varepsilon}D'_n . \tag{23} \]

The relation obtained makes it possible to find the limit of the permissible mismatch in the schedules of the district heat and electric loads.

If the quantity of steam used in the heat network \((D'_i)\) is greater than

\[ \frac{\varepsilon - 1}{\varepsilon}D'_n, \]

then installation of a back-pressure turbine leads to a definite saving of heat.

If \(D'_i\) is less than

\[ \frac{\varepsilon - 1}{\varepsilon}D'_n, \]

then heat saving will be achieved by using a separate installation, which in this case becomes more advantageous from the heat-engineering standpoint.

Using formula (22 or \(22'\)), we are able, without carrying out any other calculations, to establish whether the back-pressure-turbine variant will be possible from the heat-engineering standpoint for a given relation between the heat and electric loads.

It is simplest to show that a comparatively large mismatch of both types of load is permissible here by numerical examples. These same numerical examples will also reveal the influence of individual factors, in particular the influence of the principal factor—the magnitude of the back pressure.

Example 1.

Determine up to what value

\[ \varepsilon = \frac{D_n}{D_l} \]

the back-pressure variant of energy supply will be more advantageous than the separate-energy-supply variant, under the condition that the following units are being compared (power \(N_e\) of the order of 3000–5000 kW).

1) Condensing turbine (without regeneration)

\[ p_0 = 29 \text{ ata}; \quad p_k = 0.04 \text{ ata}; \quad t_0 = 400^\circ\mathrm{C}; \quad r_1 = 0.08, \]

\[ \eta_{0e} = 0.75; \quad \eta_{mk} = 0.98; \quad \eta_{rk} = 0.94; \quad \eta_{0ik} = 0.765 \]

2) Back-pressure turbine

\[ p_0 = 29\ \text{ata};\quad p_{\text{p}} = 1.2\ \text{ata};\quad t_0 = 400^\circ\text{C};\quad r_2 = 0.06 \]

\[ \eta_{\text{oe}} = 0.73;\quad \eta_{\text{mp}} = 0.98;\quad \eta_{\text{gp}} = 0.94;\quad \eta_{\text{oin}} = 0.745. \]

Using the \(i\)—\(s\) diagram and the tables for water vapor, we find the quantities needed for substitution into formula (22″). Formula (22″) may be used because the condition holds

\[ \eta_{\text{gk}} = \eta_{\text{gp}}, \]

\[ \eta_{\text{mk}} = \eta_{\text{mp}}. \]

1) For the condensing turbine

\[ H_0 = i_0 - i_{1\text{tk}} = 275\ \text{kcal/kg}; \]

\[ i_0 - i_{\text{k}} = H_0 \eta_{\text{oik}} = 275 \cdot 0.765 = 219\ \text{kcal/kg}; \]

\[ i_0 = 774\ \text{kcal/kg};\quad i_{\text{k}} = 774 - 210 = 564\ \text{kcal/kg}; \]

\[ i'_2 = 29\ \text{kcal/kg}. \]

2) For the back-pressure turbine

\[ H_0 = i_0 - i_{1\text{tp}} = 774 - 609 = 165\ \text{kcal/kg}; \]

\[ i_0 - i_{\text{p}} = 165 \cdot 0.745 = 123\ \text{kcal/kg}; \]

\[ i_{\text{p}} = 774 - 123 = 651\ \text{kcal/kg}; \]

\[ i_2 = 104\ \text{kcal/kg}. \]

We find \(\varepsilon\)

\[ \varepsilon = \frac{(651 - 104)(1 + 0.06)} {\dfrac{123}{210}\left[564 - 29 + 0.08(774 - 29)\right] - 0.06(774 - 651)} = 1.70 \]

By formula (23) we find

\[ D'_{\text{k}} = \frac{1.70 - 1}{1.70}D'_{\text{p}} = 0.411D'_{\text{p}}, \]

therefore, in the present case, when no less than \(41.1\%\) of the total amount of steam passing through the turbine is used to cover the district electrical load, the back-pressure installation will be more advantageous than the separate power-supply variant.

Example 2. Compare the separate power-supply variant (the same condensing turbine) with the back-pressure turbine variant, if steam with a pressure at the turbine outlet not of 1.2 ata, but of 6 ata, is required for heat consumption.

For the back-pressure turbine the following parameters are specified:

\[ p_0 = 29\ \text{ata};\quad t_0 = 400^\circ\text{C};\quad p_{\text{p}} = 6\ \text{ata};\quad r_2 = 0.06; \]

\[ \eta_{\text{oe}} = 0.73;\quad \eta_{\text{mp}} = 0.98;\quad \eta_{\text{gp}} = 0.94;\quad \eta_{\text{oin}} = 0.745. \]

We find

\[ H_0 = 774 - 679 = 95\ \text{kcal/kg}; \]

\[ i_0 - i_{\text{p}} = 95 \cdot 0.745 = 71\ \text{kcal/kg}; \]

\[ i_{\text{p}} = 774 - 71 = 703\ \text{kcal/kg}; \]

\[ i_2 = 159\ \text{kcal/kg}. \]

Since

\[ \eta_{\text{mp}} = \eta_{\text{mk}}, \]

\[ \eta_{\text{gp}} = \eta_{\text{gk}}, \]

again use equation (22″)

\[ \varepsilon = \frac{(703-159)\cdot(1+0.06)} {\dfrac{71}{210}\left[564-29+0.08(745-29)\right]-0.06(774-703)} =2.94; \]

\[ D'_{\mathrm{i}}=\frac{2.94-1}{2.94}D'_{\mathrm{p}}=0.66D'_{\mathrm{p}}. \]

As we see, in comparison with the first example, the situation has changed very unfavorably for the back-pressure turbine. In order to be able to compete with the separate power-supply variant, the back-pressure turbine must already deliver to heat consumption no less than 66.0% of the total quantity of steam expended by it to cover the electrical load. Meanwhile, in comparison with the back-pressure turbine of the first example, the quantity of steam passed through the turbine has increased in the ratio determined by the ratio of the heat drops used for power production, i.e. by

\[ \frac{123}{71} \]

times.

Thus, the magnitude of the back pressure determined by the heat consumer plays a very substantial role in the question of comparing the economy of the considered variants of district power supply. We shall show this more clearly with the aid of numerical examples.

Example 3. Determine which of the power-supply variants—separate or from a back-pressure turbine—will be more advantageous under the condition that

\[ p_{\mathrm{p}}=1.2\ \text{ata};\quad N_{\mathrm{e}}=2900\ \text{kW};\quad W=6\ \text{Mcal/h}. \]

The turbine characteristics are the same as in Example 1.

We determine the steam consumption for the back-pressure turbine

\[ D'_{\mathrm{p}}= \frac{860N_{\mathrm{e}}(1+r_2)} {(i_0-i_{\mathrm{p}})\eta_{\mathrm{mp}}\cdot\eta_{\mathrm{gp}}} = \frac{860\cdot2900(1+0.06)} {123\cdot0.98\cdot0.94} =23200\ \text{kg/h}; \]

By means of this quantity of steam, the following heat load can be covered

\[ W'=23200(651-104)\cdot10^{-6}=12.7\ \text{Mcal/h}. \]

As we see, the turbine supplies more steam than is required. The quantity of excess steam

\[ D'_{\mathrm{l}}=\frac{(12.7-6)10^6}{(651-104)}=12250\ \text{kg/h}, \]

the ratio

\[ \varepsilon_{\mathrm{fact}}=\frac{D'_{\mathrm{a}}}{D'_{\mathrm{l}}} =\frac{23200}{12250}=1.9. \]

In Example 1 the critical value of \(\varepsilon\) was determined as

\[ \varepsilon=1.70. \]

Since \(\varepsilon_{\mathrm{fact}}>\varepsilon\), the back-pressure turbine variant in the concrete case under consideration will be more advantageous than separate power supply, although 52% of the steam that has passed through the turbine (12.5 t/h!) is not used.

Example 4. Suppose that, as in Example 3, \(N_{\mathrm{e}}=2900\ \text{kW}\); \(W=6\ \text{Mcal/h}\), but the consumer requires steam not at a pressure of 1.2 ata, but at \(p_{\mathrm{p}}=6\) ata.

Solve the same problem as in Example 3, taking the turbine characteristics from Example 2.

We find

\[ D'_p=\frac{2900\cdot 860\cdot 1,06}{71\cdot 0,94\cdot 0,98}=40200\ \text{kg/hour}; \]

\[ W'=40200(703-158)\cdot 10^{-6}=21,95\ \text{Mcal/hour}. \]

It is required to cover the heat load \(W=6\ \text{Mcal/hour}\).

\[ D'_i=\frac{(21,95-6)10^6}{703-159}=29300\ \text{kg/hour}, \]

\[ \varepsilon_{\text{fact}}=\frac{40200}{29300}=1,37. \]

In Example 2 the critical value of \(\varepsilon\) was determined as \(\varepsilon=2,94\). Since \(\varepsilon_{\text{fact}}<\varepsilon\), the back-pressure turbine variant is uneconomical. As is seen from Examples 3, 4 and 1, 2, the magnitude of the back pressure, for given initial parameters, plays a large role in determining the critical value of \(\varepsilon\).

In order to trace the influence of \(p_p\) on \(\varepsilon\) and \(\dfrac{\varepsilon-1}{\varepsilon}\), we constructed the graphs shown in Fig. 6 and Fig. 7. On each of the graphs four curves are plotted, numbered with Roman numerals. In constructing

Back pressure

Fig. 6

the curves, the figures indicated in numerical Example 1 were taken as the characteristics of the condensing turbine. For the back-pressure turbine, irrespective of the back-pressure height, it was assumed that

\[ \eta_{oin}=0,745;\quad \eta_{mn}=\eta_{mk}=0,98; \]

\[ \eta_{gn}=\eta_{gk}=0,94;\quad r_2=r_1-0,02=0,06. \]

Curves I are constructed by formula 22; curves II are constructed by the same formula, but simplified by the assumption that the influence of the plant’s auxiliary requirements is insignificant. Accordingly, in the numerator and denominator of the right-hand side of formula \((22')\) the terms containing \(r_1, r_2\),

and, consequently, the points of curve II, \(\varepsilon=f(p_n)\), were calculated by using the equation

\[ \varepsilon = \frac{i_n-i_2} {\dfrac{i_0-i_n}{i_0-i_k}\left(i_k-i'_2\right)} . \]

Curves III were constructed according to the formula

\[ \varepsilon = \frac{(i_n-15)(1+r_2)} {\dfrac{i_0-i_n}{i_0-i_k}\left[\left(i_k-i'_2\right)+r_1\left(i_0-i'_2\right)\right]-r_2\left(i_0-i_n\right)} . \]

Here it is assumed that the condensate of the excess steam is not used, as a result of which the condensate losses must be made up by cold water having a heat content of \(15\ \text{kcal/kg}\).

Fig. 7

Finally, for constructing curves IV, the same formula was used as for curves III, but without taking into account the influence of auxiliary needs. Accordingly, in the denominator the terms containing \(r_1\), \(r_2\) were omitted.

Extension of the comparison method to more complex installations. Derivation of a general formula for \(\varepsilon\). Particular values of \(\varepsilon\)

The method considered above for comparing the separate power-supply variant with the back-pressure variant led us to formulas for determining the critical value of \(\varepsilon\) in the case when neither the condensing nor the back-pressure turbines are equipped with an installation for regenerative heating of water.

At the present time even small turbines very often operate with regeneration, not to mention large machines, which always make use of the advantages of regenerative heating of feed water. Therefore it appears of interest to apply the proposed comparison method to more complex cases, including in the consideration the influence of regeneration on the value of \(\varepsilon\).

In the concrete solution of this problem we must assume that the scheme of the regenerative installation is given. As is known, the scheme itself is calcu-

the determination of the steam and condensate flows in the regenerative feedwater-heating installation plays a certain role in the question of the thermal-engineering advantage of regeneration. The main factors are the final temperature of feedwater heating and the number of points of nonregulated steam extraction. The question of the type of heaters (mixing, surface), the question of the distribution of the drains of the heating steam, etc., are essentially secondary factors, capable of changing comparatively little, in one direction or another, the figures characterizing the advantages of regenerative feedwater heating.

Proceeding from these considerations, in what follows, when solving examples, we shall assume that a regenerative feedwater-heating arrangement with \(z\) steam-extraction points and with mixing heaters is being considered. Figure 8 shows such an arrangement with 3 extraction points for a condensing turbine. For simplicity we assume that there are no condensate (or steam) leaks, and therefore there is no need to be concerned with obtaining make-up water or with heating it. It is already clear from the preceding discussion that the introduction of these complicating factors (the latter being possible when a concrete scheme is available) cannot change anything in the method of obtaining the expression for \(\varepsilon\). On the other hand, these complicating factors have no fundamental significance.

Fig. 8

Comparing the separate variant of energy supply, in which the condensing turbine has a regenerative installation consisting of \(z\) mixing heaters, with the variant of energy supply from a back-pressure turbine, one may assume, for generality of the conclusions, that the back-pressure turbine also has a regenerative installation consisting of \(z_1\) heaters supplied with steam from intermediate points of the process.

Figure 9 shows the processes of both turbines being compared. For clarity of representation it is assumed that the condensing turbine has 3 extraction points (the heat content of the steam at the extraction points \(i_{10}, i_{20}, i_{30}\));

back-pressure turbine—2 extraction points (steam heat content at the extraction points \(i_{1n}; i_{2n}\)).

The next pair of installations that it appears interesting to compare with one another from the standpoint of their thermal-engineering advantage will be the simplest back-pressure turbine (without regenerative feedwater heating) and a turbine with steam extraction (without regeneration).

The thermal processes of both turbines are shown in Fig. 10 in the \(i\)—\(s\) diagram. The distribution of steam flows in the turbine with extraction is given in Fig. 11.

Fig. 9

Finally, one can compare with one another a back-pressure turbine and a turbine with steam extraction, taking into account the presence of regenerative feedwater heating in the scheme of each turbine.

Obviously, comparison of the last pair of installations will lead us to the most general formula for the critical value \(\varepsilon\). From this general formula all particular values can be obtained.

Let us suppose that two installations are being compared: 1) one having a turbine with back pressure, 2) one having a turbine with steam extraction.

Let both turbines have several uncontrolled extractions for regenerative feedwater heating. To obtain the most general conclusions, let us further assume that the initial steam parameters of the two turbines are different. In making the comparison we shall base ourselves on the same premises as before, i.e., we shall assume that either of the two machines being compared must at any moment cover the heat and electric loads of the district. The consideration also remains in force that electro-

energy to meet the station’s own needs is produced by the same turbogenerator, i.e., by the same process as the energy generated to cover the electrical load of the district.

The thermal processes of the turbines in the \(i\)—\(S\) diagram are shown schematically in Fig. 12. For definiteness of the representation, Fig. 12 shows that the extraction turbine has 3 extraction points for feedwater heating; the back-pressure turbine has 2 extraction points for the same purpose.

Let us introduce the following notation:

\(i_{m0}\) — heat content of steam from the uncontrolled extractions of the extraction turbine,

\(i_{mn}\) — heat content of steam from the uncontrolled extractions of the back-pressure turbine,

\(i_{n0}\) — heat content of steam from the controlled extraction,

Fig. 10

Fig. 11

\(D_k\) — steam flow rate through the extraction turbine for covering the electrical load of the district,

\(D_n\) — the same for the back-pressure turbine,

\(D_{n0}\) — steam extraction from the controlled extraction,

\(\alpha_{m0} = \dfrac{D_{m0}}{D_k}\) — relative magnitude of the uncontrolled extraction from the given point (extraction turbine),

\(\alpha_{mn} = \dfrac{D_{mn}}{D_n}\) — relative magnitude of the uncontrolled extraction from the given point (back-pressure turbine).

The remaining notation is given in Fig. 12.

To cover the (electrical) load of the district, the extraction turbine develops the power

\[ N_{\mathrm{e}}^{k} = \frac{ D_k\left[(i_0^{k}-i_k)-\sum \alpha_{m0}(i_{m0}-i_k)\right] - D_{n0}(i_{n0}-i_k) }{860} \,\eta_{iцk}\,\eta_{гk}. \tag{24} \]

For a back-pressure turbine, analogously,

\[ N_{\mathrm{e}}^{*}= \frac{D_{\mathrm{n}}\left[\left(i_{0}^{\mathrm{n}}-i_{\mathrm{n}}\right)-\sum \alpha_{\mathrm{mn}}\left(i_{\mathrm{mn}}-i_{\mathrm{n}}\right)\right]}{860} \,\eta_{\mathrm{mn}}\eta_{\mathrm{gn}}, \tag{25} \]

\[ N_{\mathrm{e}}^{*}=N_{\mathrm{e}}^{\mathrm{n}}, \]

i.e.

\[ \left\{ D_{\mathrm{k}}\left[\left(i_{0}^{\mathrm{k}}-i_{\mathrm{k}}\right)-\sum \alpha_{\mathrm{mo}}\left(i_{\mathrm{mo}}-i_{\mathrm{k}}\right)\right] - D_{\mathrm{no}}\left(i_{\mathrm{no}}-i_{\mathrm{k}}\right) \right\}\eta_{\mathrm{mk}}\eta_{\mathrm{gk}} = \]

\[ = D_{\mathrm{n}}\left[\left(i_{0}^{\mathrm{n}}-i_{\mathrm{n}}\right)-\sum \alpha_{\mathrm{mn}}\left(i_{\mathrm{mn}}-i_{\mathrm{n}}\right)\right]\eta_{\mathrm{mn}}\eta_{\mathrm{gn}}. \tag{26} \]

Fig. 12

From equation (26) we determine \(D_{\mathrm{k}}\) through \(D_{\mathrm{n}}, D_{\mathrm{no}}\)

\[ D_{\mathrm{k}} = D_{\mathrm{n}} \frac{ \left(i_{0}^{\mathrm{n}}-i_{\mathrm{n}}\right)-\sum \alpha_{\mathrm{mn}}\left(i_{\mathrm{mn}}-i_{\mathrm{n}}\right) }{ \left(i_{0}^{\mathrm{k}}-i_{\mathrm{k}}\right)-\sum \alpha_{\mathrm{mo}}\left(i_{\mathrm{mo}}-i_{\mathrm{k}}\right) } \cdot \frac{\eta_{\mathrm{mn}}\cdot\eta_{\mathrm{gn}}}{\eta_{\mathrm{mk}}\eta_{\mathrm{gk}}} + \]

\[ + D_{\mathrm{no}} \frac{ i_{\mathrm{no}}-i_{\mathrm{k}} }{ \left(i_{0}^{\mathrm{k}}-i_{\mathrm{k}}\right)-\sum \alpha_{\mathrm{mo}}\left(i_{\mathrm{mo}}-i_{\mathrm{k}}\right) }. \tag{27} \]

The sum of losses of a condensing turbine with steam extraction and regeneration

\[ \sum Q_{\text{loss}}^{\text{co}} = \left[ D_{\text{k}}\left(1-\sum \alpha_{\text{mo}}\right)-D_{\text{po}} \right](i_{\text{k}}-i_2') + \]

\[ + r_1 \left[ D_{\text{k}}\left(1-\sum \alpha_{\text{mo}}\right)-D_{\text{po}} \right](i_0^{\text{k}}-i_2') + \]

\[ + r_1\sum \alpha_{\text{mo}}(i_0^{\text{k}}-i_{\text{mo}})D_{\text{k}} + r_1D_{\text{po}}(i_0^{\text{k}}-i_{\text{po}}) + \]

\[ + (1-\eta_{\text{k}}) \left\{ D_{\text{k}} \left[ (i_0^{\text{k}}-i_{\text{k}}) - \sum \alpha_{\text{mo}}(i_{\text{mo}}-i_{\text{k}}) \right] - D_{\text{po}}(i_{\text{po}}-i_{\text{k}}) \right\}; \tag{28} \]

here

\[ \eta_{\text{mk}}\cdot \eta_{\text{rk}}=\eta_{\text{k}}. \]

The sum of losses of a back-pressure turbine with regeneration

\[ \sum Q_{\text{loss}}^{\text{p}} = D_{\text{l}}(i_{\text{p}}-i_2')(1+r_2) + r_2D_{\text{p}} \left(1-\sum \alpha_{\text{mn}}\right)(i_0^{\text{p}}-i_{\text{p}}) + \]

\[ + r_2D_{\text{p}}\sum \alpha_{\text{mn}}(i_0^{\text{p}}-i_{\text{mn}}) + \]

\[ + (1-\eta_{\text{p}})D_{\text{p}} \left[ (i_0^{\text{p}}-i_{\text{p}}) - \sum \alpha_{\text{mn}}(i_{\text{mn}}-i_{\text{p}}) \right]; \tag{29} \]

here

\[ \eta_{\text{mp}}\cdot \eta_{\text{rp}}=\eta_{\text{p}}. \]

The equivalence of the two installations being compared is determined by the condition

\[ \sum Q_{\text{loss}}^{\text{co}}=\sum Q_{\text{loss}}^{\text{p}}. \tag{29'} \]

Consequently, it is necessary to equate to one another the right-hand sides of expressions (28), (29), and from this find the critical value \(\varepsilon\), at which both installations will be equivalent from the thermotechnical point of view.

Denote

\[ \left. \begin{aligned} (i_0^{\mathrm{p}}-i_{\mathrm{p}})-\sum \alpha_{mn}(i_{mn}-i_{\mathrm{p}})&=A,\\ (i_0^{\mathrm{k}}-i_{\mathrm{k}})-\sum \alpha_{mo}(i_{mo}-i_{\mathrm{k}})&=B,\\ \sum \alpha_{mn}(i_0^{\mathrm{p}}-i_{mn})&=C,\\ \sum \alpha_{mo}(i_0^{\mathrm{k}}-i_{mo})&=F,\\ 1-\sum \alpha_{mn}&=K,\\ 1-\sum \alpha_{mo}&=L,\\ \eta_{mn}\cdot\eta_{t\mathrm{p}}&=\eta_{\mathrm{p}},\\ \eta_{mk}\cdot\eta_{t\mathrm{k}}&=\eta_{\mathrm{k}},\\ \frac{1+r_2}{1+r_1}\cdot\frac{i_{\mathrm{p}}-i_2}{i_{po}-i_2}&=M \quad \text{(see below)} . \end{aligned} \right\} \tag{30} \]

Substituting into equation (29′) the values \(\sum Q_{\text{loss}}^{ko}\), \(\sum Q_{\text{loss}}^{\mathrm{p}}\) from (28), (29), and using the notations (30), we obtain:

\[ \begin{aligned} &(LD_{\mathrm{k}}-D_{po})(i_{\mathrm{k}}-i_2') +r_1(LD_{\mathrm{k}}-D_{po})(i_0^{\mathrm{k}}-i_2') +r_1D_{\mathrm{k}}F \\ &\quad +r_1D_{po}(i_0^{\mathrm{k}}-i_{po}) +(1-\eta_{\mathrm{k}})\bigl[D_{\mathrm{k}}B-D_{po}(i_{po}-i_{\mathrm{k}})\bigr] \\ &=D_{\ell}(i_{\mathrm{p}}-i_2)(1+r_2) +r_2D_{\mathrm{p}}K(i_0^{\mathrm{p}}-i_{\mathrm{p}}) +r_2D_{\mathrm{p}}C \\ &\quad +(1-\eta_{\mathrm{p}})D_{\mathrm{p}}A \end{aligned} \tag{31} \]

Introducing the notations (30) into expression (27), we find

\[ D_{\mathrm{k}} = D_{\mathrm{p}}\frac{A}{B}\cdot\frac{\eta_{\mathrm{p}}}{\eta_{\mathrm{k}}} + D_{po}\frac{i_{po}-i_{\mathrm{k}}}{B}. \tag{32} \]

From expression (32) eliminate \(D_{po}\).

The steam going to extraction covers the same heat demand as the steam used from the back-pressure turbine. Both turbines supply to the heat consumer steam of the same pressure and receive back condensate with the same heat content \(i_2\).

This is expressed by writing

\[ (1+r_1)D_{po}(i_{po}-i_2) = (D_{\mathrm{p}}-D_{\ell})(i_{\mathrm{p}}-i_2)(1+r_2); \tag{33} \]

\[ D_{po} = (1+r_2)\frac{D_{\mathrm{p}}-D_{\ell}}{1+r_1}\cdot \frac{i_{\mathrm{p}}-i_2}{i_{po}-i_2} = M(D_{\mathrm{p}}-D_{\ell}). \tag{34} \]

Substituting into expression (32) the value of \(D_{po}\) from (34), we have

\[ D_{\mathrm{k}} = D_{\mathrm{p}}\frac{A}{B}\cdot\frac{\eta_{\mathrm{p}}}{\eta_{\mathrm{k}}} + M(D_{\mathrm{p}}-D_{\ell}). \tag{35} \]

Substitute the value \(D_k\) from (35) into equation (31), and after a series of transformations we obtain

\[ \begin{aligned} D_p \frac{A}{B}\left\{\frac{\eta_p}{\eta_k} \left[L(i_k-i_2')+r_1L(i_0^k-i_2')+r_1F+B\right]-B\right\}+{}\\ {}+D_pM\left\{\left(L\frac{i_{p0}-i_k}{B}-1\right) \left[(i_k-i_2')+r_1(i_0^k-i_2')\right] +r_1\left[\frac{i_{p0}-i_k}{B}F+(i_0^k-i_{p0})\right]\right\}-{}\\ {}-D_pr_2\left[K(i_0^p-i_p)+C\right] = D_l(i_p-i_2)(1+r_2)+{}\\ {}+D_lM\left\{\left(L\frac{i_{p0}-i_k}{B}-1\right) \left[(i_k-i_2')+r_1(i_0^k-i_2')\right] +r_1\left[\frac{i_{p0}-i_k}{B}F+(i_0^k-i_{p0})\right]\right\}. \end{aligned} \]

Denote

\[ \psi = M\left\{\left(L\frac{i_{p0}-i_k}{B}-1\right) \left[(i_k-i_2')+r_1(i_0^k-i_2')\right] +r_1\left[\frac{i_{p0}-i_k}{B}F+(i_0^k-i_{p0})\right]\right\} \tag{36} \]

\[ \varepsilon=\frac{D_p}{D_l}= \]

\[ = \frac{(i_p-i_2)(1+r_2)+\psi} {\dfrac{A}{B}\left\{\dfrac{\eta_p}{\eta_k} \left[L(i_k-i_2')+r_1L(i_0^k-i_2')+r_1F+B\right]-B\right\} -r_2\left[K(i_0^p-i_p)+C\right]+\psi}. \tag{37} \]

In formula (37) we carry out the following transformations:

\[ L(i_k-i_2')+B = (i_0^k-i_2')-\sum \alpha_{m0}(i_{m0}-i_2'); \]

\[ r_1L(i_0^k-i_2)+r_1F = r_1\left[(i_0^k-i_2')-\sum \alpha_{m0}(i_{m0}-i_2')\right]; \]

\[ r_2\left[K(i_0^p-i_p)+C\right] = r_2\left[(i_0^p-i_p)-\sum \alpha_{mn}(i_{mn}-i_p)\right]. \]

Consequently, the expression standing in the square brackets of the denominator of (37),

\[ [L(i_k-i_2')+B+r_1L(i_0^k-i_2')+r_1F] = \]

\[ = \left[(i_0^k-i_2')-\sum \alpha_{m0}(i_{m0}-i_2')\right](1+r_1). \]

Having denoted

\[ \sum \alpha_{m0}(i_{m0}-i_2')=R, \tag{38} \]

we can write the expression for \(\varepsilon\) in the form

\[ \varepsilon= \frac{(i_p-i_2)(1+r_2)+\psi} {\dfrac{A}{B}\left\{\dfrac{\eta_p}{\eta_k} \left[(i_0^k-i_2')-R\right](1+r_1)-B\right\} -r_2A+\psi}, \tag{39} \]

where, for brevity, the designations (30), (36), and (38) have been used.

By the simplest transformations of the denominator of formula (39) we obtain

\[ \varepsilon = \frac{(i_{\mathrm n}-i_2)(1+r_2)+\mu} {\dfrac{A}{B}\cdot \dfrac{\eta_{\mathrm n}}{\eta_{\mathrm k}} \left[(i_0^{*}-i_2')-R\right](1+r_1)-A(1+r_2)+\mu}. \tag{40} \]

From formulas (39), (40) one can obtain particular formulas for the critical value of \(\varepsilon\) in the case where two simpler installations are being compared. Let us show this.

Suppose, for example, that variants of power supply from a back-pressure turbine and from a turbine with steam extraction are being compared. The initial steam parameters in both variants are identical. Both turbines operate without regenerative water heating (Fig. 10).

In this case formula (39) can easily be reduced to the form

\[ \varepsilon = \frac{(i_{\mathrm n}-i_2)(1+r_2)+\mu} {\dfrac{i_0-i_{\mathrm n}}{i_0-i_{\mathrm k}} \left[ \dfrac{\eta_{\mathrm{mn}}\eta_{\mathrm{gn}}}{\eta_{\mathrm{mk}}\eta_{\mathrm gk}} (i_0-i_2')(1+r_1)-(i_0-i_{\mathrm k}) \right] -r_2(i_0-i_{\mathrm n})+\mu}, \tag{41} \]

because

\[ i_0^{\mathrm k}=i_0^{\mathrm n}=i_0, \]

\[ A=i_0-i_{\mathrm n}, \]

\[ B=i_0-i_{\mathrm k}, \]

\[ R=0, \]

since there are no uncontrolled steam extractions.

In the present case:

\[ \mu = M\left\{ \left(\frac{i_{\mathrm{no}}-i_{\mathrm k}}{i_0-i_{\mathrm k}}-1\right) \left[(i_{\mathrm k}-i_2')+r_1(i_0-i_2')\right] +r_1(i_0-i_{\mathrm{no}}) \right\}, \]

since

\[ F=0, \]

\[ L=1, \]

From formula (41) it is easy to obtain the formula (22) already known to us.

If there is no steam extraction, i.e., if the variants considered in deriving formula (22) are being compared, then

\[ D_{\mathrm{no}}=0. \]

It follows from (34) that in this case also

\[ M=\frac{D_{\mathrm{no}}}{D_{\mathrm n}-D_{\mathrm l}}=0. \]

Consequently, \(\mu=0\).

Thus, in the present case formula (41) is transformed into formula (22).

The same conclusion could have been reached by taking formula (39) directly as the starting point.

We refrain from giving particular formulas for \(\varepsilon\) obtained when comparing different combinations of power-supply variants, since in calculations it is much simpler to use the general formula (39) or (40), computing the values of the quantities \(A, B, R, \mu\) entering into it for the particular pair of variants under consideration.

In order to establish the competitiveness of a back-pressure turbine as compared with a turbine having intermediate ex—

steam extraction, let us examine formula (41), written for the case when each of the two installations being compared has no regenerative water heating in its scheme, i.e., is the simplest one.

It goes without saying that a back-pressure turbine can compete more easily with a turbine of the purely condensing type than with a turbine having intermediate steam extraction. In accordance with what has been said, the critical value

\[ \frac{D_{\mathrm{p}}}{D_{\mathrm{l}}}=\varepsilon \]

in the second case should be greater than in the first. Let us see how this agrees with formulas (41) and (22).

Let us consider the polynomial expression which appears both in the numerator and in the denominator and is denoted by \(\mu\)—

\[ \mu=M\left\{\left(\frac{i_{\mathrm{po}}-i_{\mathrm{k}}}{i_0-i_{\mathrm{k}}}-1\right) \left[(i_{\mathrm{k}}-i_2')+r_1(i_0-i_2')\right]+r_1(i_0-i_{\mathrm{po}})\right\}. \]

It is not difficult to show that, when calculating this expression, we shall always obtain a negative quantity; that is, both the numerator and the denominator of expression (41) will be smaller by this quantity than the numerator and denominator of formula (22). Consequently, in calculations of \(\varepsilon\) by formula (41) we shall obtain larger values than when calculating this quantity by formula (22).

To show how hopeless competition with a turbine having steam extraction is for a back-pressure turbine, let us solve two numerical examples, using formula (41).

Example 5. Determine the critical value of \(\varepsilon\) when comparing two installations. One of them has a back-pressure turbine characterized by the figures of example 1; the other is represented by a turbine with steam extraction; the process of this turbine is shown schematically in Fig. 10. In plotting the process it is assumed that the high-pressure cylinder has \(\eta_{oi}=0.69\); the low-pressure cylinder has \(\eta_{oi}=0.80\); the initial parameters are the same as for the back-pressure turbine \((p_0=29\ \text{ata};\ t_0=400^\circ\mathrm{C})\); the pressure in the condenser is \(p_{\mathrm{k}}=0.04\ \text{ata}\); steam extraction is carried out at a pressure of \(1.2\ \text{ata}\); \(r_1=r_2=0.06\). The mechanical efficiencies and generator efficiencies of both turbines are the same.

For the solution of the problem we take formula (41).

From the data of example 1 (for the back-pressure turbine) and from the process of the turbine with steam extraction, we find:

\[ i_{\mathrm{n}}-i_2=651-104=547\ \text{kcal/kg}, \]

\[ i_{\mathrm{po}}-i_2=660-104=556\ \text{kcal/kg}, \]

\[ i_{\mathrm{po}}-i_{\mathrm{k}}=660-568=92\ \text{kcal/kg}, \]

\[ i_0-i_{\mathrm{k}}=774-568=206\ \text{kcal/kg}, \]

\[ i_{\mathrm{k}}-i_2'=568-29=539\ \text{kcal/kg}, \]

\[ i_0-i_2'=774-29=745\ \text{kcal/kg}, \]

\[ i_0-i_{\mathrm{po}}=774-660=114\ \text{kcal/kg}, \]

\[ i_0-i_{\mathrm{n}}=774-651=123\ \text{kcal/kg}. \]

We compute the value of \(\mu\)

\[ \mu=\frac{1+0.06}{1+0.06}\cdot\frac{547}{556} \left[ \left(\frac{92}{206}-1\right)(539+0.06\cdot745)+0.06\cdot114 \right]=-311; \]

\[ \varepsilon= \frac{547(1+0.06)-311} {\frac{123}{206}\left[(1+0.06)\cdot745-206\right]-0.06\cdot123-311} \simeq 8.8. \]

Thus, in order that the back-pressure turbine be equivalent to the turbine with steam extraction, under the specified concrete process conditions, it is necessary to use

\[ \frac{\varepsilon - 1}{\varepsilon} = \frac{8.8 - 1}{8.8} = 0.887 \]

of the amount of steam entering the back-pressure turbine to cover the electrical load.

As we see, despite the cumbersome appearance of formula (41), calculations by it are extremely simple and take very little time.

Example 6. Determine the critical value of \(\varepsilon\) when comparing two installations. One of them has a back-pressure turbine with the same \(\eta_{oi}\) as in Example 2 (\(\eta_{oi}=0.745\)); the other is represented by a turbine with steam extraction; the initial parameters of both turbines are identical: \(p_0 = 29\) ata; \(t_0 = 400^\circ\mathrm{C}\). The back pressure is 6 ata; steam extraction is at the same pressure. The pressure in the condenser of the turbine with steam extraction is \(p_k = 0.04\) ata; \(r_1 = r_2 = 0.06\).

The process of the turbine with steam extraction is shown schematically in Fig. 10. In this case

\[ i_{no}=709\ \text{kcal/kg};\quad i_k=558\ \text{kcal/kg};\quad i_2'=29\ \text{kcal/kg}, \]

\[ i_0=774\ \text{kcal/kg};\quad i_2=159\ \text{kcal/kg}. \]

For the back-pressure turbine,

\[ i_0=774\ \text{kcal/kg};\quad i_p=703\ \text{kcal/kg}. \]

We determine the quantities required by us:

\[ i_p-i_2=703-159=544\ \text{kcal/kg}, \]

\[ i_{no}-i_2=709-159=550\ \text{kcal/kg}, \]

\[ i_{no}-i_k=709-558=151\ \text{kcal/kg}, \]

\[ i_0-i_k=774-558=216\ \text{kcal/kg}, \]

\[ i_0-i_2'=774-29=745\ \text{kcal/kg}, \]

\[ i_k-i_2'=558-29=529\ \text{kcal/kg}, \]

\[ i_0-i_{no}=774-709=65\ \text{kcal/kg}, \]

\[ i_0-i_p=774-703=71\ \text{kcal/kg}. \]

Regeneration is absent.

\[ \psi = \frac{1+0.06}{1+0.06}\cdot \frac{544}{550} \left[ \left( \frac{151}{216}-1 \right) (529+0.06\cdot745) + 0.06\cdot65 \right] = -167; \]

\[ \varepsilon = \frac{544(1+0.06)-167} {\dfrac{71}{216}\left[(1+0.06)745-216\right]-0.06\cdot71-167} = 23.2. \]

Consequently, in order to be able to compete with a turbine having steam extraction, the back-pressure turbine must be placed under such conditions that

\[ D_n' = \frac{\varepsilon-1}{\varepsilon}D_n' = \frac{23.2-1}{23.2}D_n' = 0.965\,D_n', \]

i.e., for the needs of heat consumption, no less than 96.5% of the total amount of steam that has passed through the turbine must be used.

The conclusion that we must draw from comparing the figures obtained as a result of solving Examples 5 and 6 is quite natural: the higher the pressure of the steam consumed by the heat users, the more difficult it is for the back-pressure turbine to compete with the turbine with steam extraction.

The second conclusion may be still more definite. Only in exceptionally rare cases—practically when the heat consumption is greater than that required for the generation of all the electric energy—is a back-pressure turbine more advantageous than a turbine with steam extraction.

Thus, already in the example of comparing the simplest installation with steam extraction and a back-pressure installation, the noncompetitiveness of the back-pressure turbine has become apparent. Therefore there is no sense in comparing installations of this type complicated by regenerative heating.

Let us see how the presence of regenerative feed-water heating affects the critical value \(\varepsilon\) when comparing the alternatives of separate power supply and power supply from a back-pressure turbine; for this purpose we shall turn to a numerical example.

Example 7. Determine which of the alternatives—separate power supply or power supply from a back-pressure turbine—will be more advantageous, taking the figures characterizing the electric and heat load from Example 3.

The changes introduced into the conditions of this example consist in the fact that the condensing turbine operates with regeneration according to the scheme of Fig. 8. The ideal case is considered, in which there are no condensate losses.

The condensing turbine is characterized by the following figures:

\[ p_0 = 29\ \text{ata};\quad t_0 = 400^\circ\text{C};\quad \eta_{oi k}=0.765;\quad r_1=0.08; \]

\[ i_0=774\ \text{kcal/kg};\quad \eta_{gk}=0.94;\quad \eta_{mk}=0.98; \]

steam extraction for water heating is carried out at three points at \(4.5\ \text{ata}\), \(1.2\ \text{ata}\), \(0.3\ \text{ata}\); correspondingly, plotting the process on the \(i\)—\(s\) diagram, we find

\[ i_{10}=700\ \text{kcal/kg};\quad i_{20}=655\ \text{kcal/kg};\quad i_{30}=615\ \text{kcal/kg}; \]

\[ i_{\mathrm{I}}=148\ \text{kcal/kg};\quad i_{\mathrm{II}}=104\ \text{kcal/kg};\quad i_{\mathrm{III}}=68.6\ \text{kcal/kg}. \]

For the back-pressure turbine the characteristics are the same as in the preceding examples.

To determine the critical value \(\varepsilon\), we use formula (39), setting in it, as specified, \(\mu=0\) and \(\dfrac{\eta_p}{\eta_k}=1\).

We determine the quantities entering into this formula:

\[ \alpha_{10}=\frac{i_{\mathrm{I}}-i_{\mathrm{II}}}{i_{10}-i_{\mathrm{II}}} =\frac{148-104}{700-104} =0.0735\ \text{kg/kg}; \]

\[ \alpha_{20}= \frac{(1-\alpha_{10})(i_{\mathrm{II}}-i_{\mathrm{III}})}{i_{20}-i_{\mathrm{III}}} = \frac{(1-0.0735)(104-68.6)}{655-68.6} =0.0565\ \text{kg/kg}; \]

\[ \alpha_{30}= \frac{(1-\alpha_{10}-\alpha_{20})(i_{\mathrm{III}}-i_2')}{i_{30}-i_2'} = \frac{(1-0.0735-0.0565)(68.6-29)}{615-29} =0.0584\ \text{kg/kg}. \]

From the preceding examples we have

\[ i_1-i_2=651-104=547\ \text{kcal/kg};\quad A=774-651=123\ \text{kcal/kg};\quad i_0-i_k= \]

\[ =774-564=210\ \text{kcal/kg}. \]

On the basis of the available figures we find:

\[ \begin{aligned} \alpha_{10}(i_{10}-i_k)&=0.0734(700-564)=10\ \text{kcal/kg};\\ \alpha_{20}(i_{20}-i_k)&=0.0565(655-564)=5.15\ \text{kcal/kg};\\ \alpha_{30}(i_{30}-i_k)&=0.0584(615-564)=2.99\ \text{kcal/kg};\\ \sum \alpha_{m0}(i_{m0}-i_k)&=18.14\ \text{kcal/kg}; \end{aligned} \]

\[ B=(i_0-i_k)-\sum \alpha_{m0}(i_{m0}-i_k)=210-18.14=191.86 \simeq 192\ \text{kcal/kg}; \]

\[ \begin{aligned} \alpha_{10}(i_{10}-i_2')&=0.0735(700-29)=49.3\ \text{kcal/kg};\\ \alpha_{20}(i_{20}-i_2')&=0.0565(655-29)=35.4\ \text{kcal/kg};\\ \alpha_{30}(i_{30}-i_2')&=0.0584(616-29)=34.2\ \text{kcal/kg};\\ \sum \alpha_{m0}(i_{m0}-i_2')&=R=118.9\ \text{kcal/kg};\\ (i_0-i_2')-R&=774-29-118.9=626.1\ \text{kcal/kg}; \end{aligned} \]

\[ \varepsilon= \frac{547\cdot 1.06} {\dfrac{123}{192}\left[626(1+0.08)-192\right]-0.06\cdot 123} =1.92. \]

From example 3 we see that the actual ratio is

\[ \varepsilon_{\text{act}}=\frac{D_n'}{D_n}=1.9. \]

Since \(\varepsilon_{\text{act}}<\varepsilon\), the back-pressure-turbine variant is thermotechnically disadvantageous. True, the difference between \(\varepsilon_{\text{act}}\) and \(\varepsilon\) is such that, in practice, both variants may be considered equivalent from the thermotechnical point of view. In this case about 12 t/h of steam leaving the

Back pressure
Fig. 13

back-pressure turbine (with a total hourly flow rate of \(\sim 23\) t/h) will not be used.

As was shown above, the critical value of \(\varepsilon\) depends to a considerable degree on the magnitude of the back pressure in the back-pressure turbine.

In Figs. 13, 14 are given the dependences \(\varepsilon=f(p_n)\) and \(\dfrac{\varepsilon-1}{\varepsilon}=\varphi(p_n)\). In constructing these curves, marked by the letter A, the characteristics of the condensa-

of the condensing turbine are taken from example 7, i.e., allowance is made for the circumstance that the condensing turbine operates with regeneration. For the back-pressure turbine the same conditions were taken as for the construction of curves

Fig. 14

I in Figs. 6, 7. In particular, it is assumed that the efficiency of the turbine does not depend on the magnitude of the back pressure.

In order to represent more clearly the influence of regeneration of the condensing turbine on \(\varepsilon\), \(\dfrac{\varepsilon - 1}{\varepsilon}\), the curves I from Figs. 6, 7 have been transferred to Figs. 13, 14. These curves are marked with the letter B.

General expression of efficiency for various heat-power installations

The analytical investigations carried out in the preceding sections make it possible for us to propose our solution with respect to the technical index of a heat-power installation, characterizing it from the point of view of economy in heat consumption, i.e., ultimately, fuel consumption.

Taking any interval of time whatever (hour, day, month, year), we can, for a purely power installation, write

\[ \eta_{\mathrm{e}} = \frac{860\,\mathrm{E}}{Q_{\mathrm{t}}}; \tag{42} \]

where \(\mathrm{E}\) is the production of electrical (mechanical) energy in kWh during the period of time under consideration;

\(Q_{\mathrm{t}}\) is the heat contained in the fuel burned under the boilers (in the case of steam engines) or in the engine cylinders during the same interval of time.

\(\eta_{\mathrm{e}}\) is the economic efficiency of the power installation.

For the economic efficiency of a heat-power installation it appears entirely logical to write

\[ \eta_{\mathrm{e}} = \frac{860\,\mathrm{E} + W}{Q_{\mathrm{t}}}, \tag{43} \]

where \(E, Q_T\) have the same meaning as in formula (42), and \(W\) is the quantity of heat used by the heat consumers over the same interval of time.

In recent years the question of a formula for determining the efficiency of heat-power installations has been the object of very many proposals concerning the structure of the formula for efficiency [8—11]. Alongside this, work has been carried out to create formulas for determining the specific fuel consumption per delivered megacalorie and per delivered kilowatt-hour both for individual heat-power installations [8, 10] and for complex electric-power systems [12]. The path to formulas for determining the specific fuel consumption per megacalorie or kilowatt-hour again lies through the efficiency of the installation.

We shall at once define our position on this question. In our opinion, Prof. I. N. Butakov is undoubtedly right in defending the necessity of applying the above formula (43) to any heat-power installation, however complex it may be in its structure.

And Prof. M. M. Khazen is undoubtedly wrong in proposing in his work [11] a new method for evaluating the perfection of steam-power installations by comparing them with a “model” installation having definite and strictly specified operating conditions. The path of studying a process by comparing this process with a specially constructed ideal cycle is not, in itself, anything unusual. This method is widely used by technical thermodynamics. From this standpoint we would have no right to reproach Prof. M. M. Khazen. But the path chosen by him is unsuccessful and does not lead to success. On the pages of our periodical press there have already been published remarks by individual representatives of Soviet science [8, 13], drawing attention to the vagueness and confusion of M. M. Khazen’s work, which at times lead to his own rejection of the very definitions he has given.

We reject Khazen’s method entirely, and if we dwell on it, we do so only because we wish to use certain propositions of Khazen—incorrect propositions—to contrast with our premises for constructing a general expression for the efficiency of heat-power installations.

Neither Prof. Butakov nor Prof. Yanovsky carried the criticism of Khazen’s work through to the end. Meanwhile, to carry it through to the end is not difficult. For this it is enough to show that Khazen makes an error already in his premises. Revealing this error will help us understand our views on the question of efficiency. It is from these latter considerations that we find it necessary to dwell on M. M. Khazen’s initial error.

Having rejected formula (43) as unsuitable, Khazen writes the following:

“For obtaining an index reflecting the actual operating conditions, it is necessary to take into account that, of two cogeneration installations with back pressure, the more perfect is the one which, with equal heat expenditures and equal delivery of thermal energy for the needs of heat consumption, will give a larger quantity of electrical energy, and at the same time to take into account the influence both of the steam parameters and of the technical perfection of the equipment.”

Let us try to illustrate graphically what Khazen has said (Fig. 15). By virtue of what can a larger quantity of electrical energy be obtained in one of the installations being compared? The same quantity of heat is supplied to both installations. In both cases the same quantity of heat is transferred to the heat consumer (i.e., is contained in the exhaust steam). Evidently, equal quantities of heat remain for the generation of electrical energy. Consequently, the internal work of the turbines in both cases must be the same. No other conclusion is possible.

Consequently, if one of the two installations being compared generates a larger quantity of electrical energy, this is possible only because it has smaller electrical and mechanical loss-

Figure labels: “Remains for production, 23”; “Heat output”; “Heat of the consumer”; “Heat consumption of the installation”; “I.”; “II.”

Fig. 15

Thus, under the formulation of the conditions given by Khazen, the criterion of the perfection of the installation will be the product $\eta_{\mathrm{m}}\eta_{\mathrm{g}}$.

Indeed, from each kilogram of steam, for the generation of electrical energy there is expended

\[ (i_0 - i_{1t})\eta_{0i}\ \text{cal/kg}, \]

and converted into electrical energy

\[ (i_0 - i_{1t})\eta_{0i}\cdot \eta_{\mathrm{m}}\eta_{\mathrm{g}}, \]

and, therefore, the ratio

\[ \eta = \frac{(i_0 - i_{1t})\eta_{0i}\eta_{\mathrm{m}}\eta_{\mathrm{g}}} {(i_0 - i_{1t})\eta_{0i}} = \eta_{\mathrm{m}}\eta_{\mathrm{g}} \tag{44} \]

characterizes the perfection of the back-pressure installation. But Khazen himself objects to this, calling expression (44) unfounded and irrational.

We have no disagreement with Khazen on the latter judgment. However, it seems to us that Prof. Khazen contradicts himself. Having incorrectly formulated the initial premises, he, of course, cannot obtain a correct solution of the problem.

Khazen’s very posing of the question is incorrect. Since a steam-power installation serves a definite group of consumers, for it there exists a definite ratio between the amount of heat expended to cover the electrical load and the amount of heat expended to satisfy the heat demand. Let us put in its place another steam-power installation. The conditions of operation will not change from this, but the economy of operation in the two cases will, generally speaking, be different.

Taking the foregoing into account, the problem of finding an index reflecting the economy of operation of a heat-power installation may be formulated in the following words: of several district-heating installations, the more perfect is that which, cover-

…covering the specified heat consumption and providing the specified electric output, expends a smaller amount of heat.

Under the given specific conditions, some quite definite installation must prove to be the most advantageous, the type and parameters of which are determined mainly by the specific operating conditions, i.e., by the degree of linkage between the heat and electric loads. It seems to us that our formulation is correct. It, of course, contains nothing new. In essence it is a reflection of formula (43); nevertheless, we consider it necessary to explain it.

Let us first consider the steam-power installation only in the part directly connected with the delivery of electric and heat energy to the consumer, i.e., the machine room.

Assessing the perfection of an installation in accordance with our formulation, and for the time being not taking into account the losses of the boiler plant and the steam line, we can illustrate our ideas with the aid of Fig. 16. This figure shows the distribution of heat in various heat-power installations placed under identical conditions from the point of view of the need to cover identical heat and electric loads. The heat expended to obtain the heat carrier at the initial parameters is distributed among the following items of expenditure:

Fig. 16

1. Heat delivered to cover the heat consumption \(W\).
2. Heat converted into electric energy (to cover the electric load of the district) \(860N_e\).
3. Specific losses of the installation, including also the electrical auxiliary consumption.

Whatever the paths by which the heat consumption and the electric load of the district are covered, the amount of heat used for these purposes will not change, cannot change. This does not depend on the designer of the station or on the designer of the machine. This amount of heat is determined only by the presence of a definite group of consumers, thermal and electrical. The specific losses of the installation, on the contrary, are determined by the type of equipment selected (we are speaking for the time being of the machine room). The correspondence of the selected type of machine to the specific operating conditions has a direct effect also on the specific losses of the installation.

In a condensing installation such specific losses will be mainly:

1) loss in the condenser;
2) loss for supplying the plant’s own needs;
3) mechanical and electrical losses.

For a back-pressure installation the specific losses are made up of the following principal items:

1) loss with surplus steam that has passed through the turbine but cannot be used because of an insufficient heat load;
2) loss for supplying the plant’s own needs;
3) mechanical and electrical losses.

Will the formulation of the question given above be correct? It appears that it is. After all, it is necessary to supply the district with heat and electric power while expending the minimum possible amount of fuel. Under the latter condition, from the thermotechnical point of view, an optimum will be obtained.

The concrete conditions determine the type of installation. The concrete conditions also determine its economy.

Suppose that any number of advantageous theoretical premises are taken as the basis for constructing a unit, but that the concrete operating conditions do not suit this type of installation. The discrepancy will immediately be reflected in the specific losses of the installation and will lead to a deterioration of the technical index characterizing the degree of heat utilization. Consequently, on the basis of the considerations set forth, the technical index of the perfection of an installation must be composed according to the following principle:

\[ \eta_{\mathrm{t}}=\frac{Q_{1}+Q_{2}}{Q_{1}+Q_{2}+Q_{3}}, \tag{45} \]

or, in a more complete form,

\[ \eta_{\mathrm{t}}=\frac{\Sigma Q_{\mathrm{pol}}}{\Sigma Q_{\mathrm{pol}}+\Sigma Q_{\mathrm{pot}}}. \tag{46} \]

We do not in any way limit the number of terms in the numerator and denominator of expression (46). This makes it possible, by formula (46), to take into account not only the perfection of the operation of an individual machine or installation, but also the simultaneous operation of as large as desired a number of the most varied installations serving the district of consumption under consideration. It is only necessary each time, having introduced an addition into the numerator, to introduce the corresponding addition into the denominator of formula (46).

Let us now turn to consideration of the steam-power installation as a whole. For the economic efficiency of such an installation as a whole the formula is written

\[ \eta_{\mathrm{e}}=\eta_{\mathrm{t}}\cdot\eta_{\mathrm{p}}\cdot\eta_{\mathrm{ku}}, \tag{47} \]

where \(\eta_{\mathrm{t}}\) is the efficiency of the engine room,
\(\eta_{\mathrm{p}}\) is the efficiency of the steam pipeline,
\(\eta_{\mathrm{ku}}\) is the efficiency of the boiler plant.

Formula (47) is quite convenient to use. Therefore nothing can be objected to in its application.

Nothing prevents us, also for the efficiency of the steam pipeline, from writing an expression for \(\eta_{\mathrm{p}}\) in the same form as for the efficiency of the engine room:

\[ \eta_{\mathrm{p}}= \frac{\Sigma Q_{\mathrm{pol}}+\Sigma Q_{\mathrm{pot}}} {\Sigma Q_{\mathrm{pol}}+\Sigma Q_{\mathrm{pot}}+\Sigma Q_{\mathrm{pot}}^{\mathrm{par}}}, \tag{48} \]

here the values \(\Sigma Q_{\mathrm{pol}}\), \(\Sigma Q_{\mathrm{pot}}\) are the same as in formula (46); \(\Sigma Q_{\mathrm{pot}}^{\mathrm{par}}\) combines all the additional losses along the path of steam transport from the boilers to the engine room.

The term \(\sum Q_{\text{loss}}^{\text{steam}}\) obviously includes:

1) losses from cooling of the steam line;
2) steam leaks from the steam line;
3) steam consumption for blowing, purging steam lines, etc.

For the boiler plant, analogously, we may write:

\[ \eta_{\text{bp}} = \frac{ \sum Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}} }{ \sum Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}} + \sum Q_{\text{loss}}^{\text{bp}} }, \]

\(\sum Q_{\text{loss}}^{\text{bp}}\) combines all losses of the boiler plant connected with the use of the heat of the fuel for steam production.

\(\sum Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}}\) is the amount of heat expended on steam production in an ideal boiler plant operating without losses, i.e. utilizing all the heat of the fuel.

It is clear that \(\sum Q_{\text{loss}}^{\text{bp}}\) is again specific to each boiler plant, just as \(\sum Q_{\text{loss}}^{\text{steam}}\) is specific to each installation.

Substitute the values \(\eta_t\), \(\eta_p\), \(\eta_{\text{bp}}\) from (46), (48), (49) into expression (47) for \(\eta_e\):

\[ \eta_e = \frac{\sum Q_{\text{use}}}{\sum Q_{\text{use}} + \sum Q_{\text{loss}}} \cdot \frac{\sum Q_{\text{use}} + \sum Q_{\text{loss}}} {\sum Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}}} \cdot \]

\[ \cdot \frac{ \sum Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}} }{ \sum Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}} + \sum Q_{\text{loss}}^{\text{bp}} }. \]

Hence, after carrying out the cancellations, we obtain

\[ \eta_e = \frac{ \sum Q_{\text{use}} }{ Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}} + \sum Q_{\text{loss}}^{\text{bp}} }, \tag{50} \]

but

\[ \sum Q_{\text{use}} = 860E + W^{1)}, \tag{51} \]

\[ \sum Q_{\text{use}} + \sum Q_{\text{loss}} + \sum Q_{\text{loss}}^{\text{steam}} + \sum Q_{\text{loss}}^{\text{bp}} = Q_t . \tag{52} \]

Therefore expression (50), upon substituting the numerator and denominator values from (51), (52), becomes

\[ \eta_e = \frac{860E + W}{Q_t}, \]

i.e. formula (43).

\(^{1)}\) When assigning the energy output to 1 hour, \(E\) should be replaced by \(N_e\).

It is interesting to note that, in the practical operation of a thermal power installation as well, the intermediate efficiencies are “contracted,” and we judge the perfection of the operation of the installation as a whole by comparing two figures: the quantity of heat utilized and the quantity of heat expended in the form of fuel, excluding the intermediate links from consideration.

And only when it is necessary to become more closely acquainted with the economy of operation of these links do we study them separately, in order to discover the link that has an unfavorable influence on the efficiency of the entire installation.

Formulas (46), (48), (49) make it possible to analyze separately $\eta_{\mathrm{t}}$, $\eta_{\mathrm{p}}$, $\eta_{\mathrm{ku}}$.

Of greatest interest from the standpoint of solving the problem we have posed—comparing the economy of various power-supply variants—is the investigation of expression (46) for $\eta_{\mathrm{t}}$.

Let us note that, in essence, almost all the authors indicated in the footnote on p. 275 confine the problem of finding a formula for the efficiency of a thermal power installation to constructing such a formula for $\eta_{\mathrm{t}}$ which would be universal and suitable for any type of thermal power installation, no matter how complex it might be. It seems to us that our formula (46) is such a most general expression for $\eta_{\mathrm{t}}$.

It is equally suitable both for an isolated unit and for any arbitrarily large number of units included in a system, since in the numerator and denominator we may sum any number of terms. On this basis it will be suitable for a turbine with extraction of steam, which represents a combination of two different types of machines (condensing and back-pressure), and for a binary installation. It also covers such installations as the Claude and Boucherot installation. For the latter type of installation, a separate investigation precisely of the expression for $\eta_{\mathrm{t}}$ is of special interest, since here there is no fuel consumption, and formula (43) in its general form is not applicable. It is also suitable for a purely power-producing installation, which represents a particular case of a district-heating installation at $W = 0$.

Let us show that this universality does indeed hold. To this end, on the basis of formula (46), let us find expressions for $\eta_{\mathrm{t}}$ in those cases where these expressions are known and do not give rise to objections.

I. Condensing installation producing only electrical energy

We shall relate our reasoning to 1 hour. We do this for the reason that in the preceding sections we considered this the most convenient from the methodological standpoint.

In the present case

\[ \Sigma Q_{\mathrm{use}} = 860 N_{\mathrm{e}}; \]

\[ \Sigma Q_{\mathrm{loss}} = r_1 D_k (i_0 - i_2') + D_k (i_k - i_2') + (1 - \eta_{\mathrm{m}}\eta_{\mathrm{g}}) D_k (i_0 - i_k); \]

\[ \eta_{\mathrm{t}} = \frac{860 N_{\mathrm{e}}} {860 N_{\mathrm{e}} + D_k (i_k - i_2') + r_1 D_k (i_0 - i_2') + D_k (1 - \eta_{\mathrm{m}}\eta_{\mathrm{g}})(i_0 - i_k)} ; \]

but

\[ D_k (i_0 - i_k)\eta_{\mathrm{m}}\eta_{\mathrm{g}} = 860 N_{\mathrm{e}}, \]

therefore

\[ \eta_{\mathrm{t}} = \frac{860 N_{\mathrm{e}}} {D_k (i_k - i_2') + D_k (i_0 - i_k) + r_1 D_k (i_0 - i_2')} . \]

\[ \eta_{\mathrm{t}} = \frac{860 N_{\mathrm{e}}} {D_k (1 + r_1)(i_0 - i_2')} \tag{53} \]

If, as is usually done, the expenditure of electrical energy for the plant’s own needs is neglected, i.e., if one puts \(r_1=0\),

\[ \eta_{\mathrm{T}}=\frac{860N_{\mathrm{e}}}{D_{\mathrm{k}}(i_0-i'_2)}, \tag{54} \]

i.e., we obtain the usual expression for the electrical efficiency of an actual condensing plant.

For an arbitrary interval of time, during which \(\mathrm{E}\) kWh have been delivered and \(D\) kg of steam have been expended, with the corresponding weighted-average values of \(i_0, i'_2, r_1\), we find:

\[ \eta_{\mathrm{T}}=\frac{860\mathrm{E}}{D(i_0-i'_2)(1+r_1)} \]

and

\[ \eta_{\mathrm{T}}=\frac{860\mathrm{E}}{D(i_0-i'_2)}. \]

II. Counterpressure plant covering heat and electric consumption according to free schedules

Here

\[ \sum Q_{\mathrm{use}}=860N_{\mathrm{e}}+W; \]

\[ \sum Q_{\mathrm{loss}}=D_{\mathrm{l}}(1+r_2)(i_{\mathrm{p}}-i_2)+r_2D_{\mathrm{p}}(i_0-i_{\mathrm{p}})+(1-\eta_{\mathrm{m}}\eta_{\mathrm{g}})D_{\mathrm{p}}(i_0-i_{\mathrm{p}}); \]

\[ \eta_{\mathrm{T}}= \frac{860N_{\mathrm{e}}+W} {860N_{\mathrm{e}}+W+D_{\mathrm{l}}(1+r_2)(i_{\mathrm{p}}-i_2)+r_2D_{\mathrm{p}}(i_0-i_{\mathrm{p}})+(1-\eta_{\mathrm{m}}\eta_{\mathrm{g}})D_{\mathrm{p}}(i_0-i_{\mathrm{p}})}. \]

But

\[ 860N_{\mathrm{e}}=D_{\mathrm{p}}(i_0-i_{\mathrm{p}})\eta_{\mathrm{m}}\eta_{\mathrm{g}}; \]

\[ W=(1+r_2)(D_{\mathrm{p}}-D_{\mathrm{l}})(i_{\mathrm{p}}-i_2); \]

substituting the values of \(860N_{\mathrm{e}}\) and \(W\) into the denominator, we find:

\[ \eta_{\mathrm{T}}= \frac{860N_{\mathrm{e}}+W} {D_{\mathrm{p}}(i_0-i_2)(1+r_2)}, \tag{55} \]

where

\[ W=(1+r_2)(D_{\mathrm{p}}-D_{\mathrm{l}})(i_{\mathrm{p}}-i_2). \]

Let us take limiting cases:

1) the heat and electrical loads are fully coordinated.
In this case

\[ D_{\mathrm{l}}=0, \]

\[ W=(1+r_2)D_{\mathrm{p}}(i_{\mathrm{p}}-i_2), \]

\[ 860N_{\mathrm{e}}=D_{\mathrm{p}}(i_0-i_{\mathrm{p}})\eta_{\mathrm{m}}\eta_{\mathrm{g}} \]

and

\[ \eta_{\mathrm{T}}= \frac{(i_0-i_{\mathrm{p}})\eta_{\mathrm{m}}\eta_{\mathrm{g}}+(i_{\mathrm{p}}-i_2)(1+r_2)} {(i_0-i_2)(1+r_2)}. \tag{55'} \]

If \(\eta_{\mathrm{m}}\eta_{\mathrm{g}}=1,\ r_2=0\), then for such an ideal plant we obtain

\[ \eta_{\mathrm{T}}=\frac{i_0-i_2}{i_0-i_2}=1; \]

2) the heat load is absent; the back-pressure turbine operates into an (atmospheric) condenser in order to preserve the feed water and the heat contained in it.

\[ D_l = D_n; \]

\[ W = 0; \]

\[ 860N_e = D_n(i_0-i_n)\eta_m\eta_g; \]

\[ \eta_t=\frac{860N_e}{D_n(i_0-i_2)(1+r_2)}. \tag{55″} \]

When operating to exhaust, \(\eta_t\) will decrease still more, since in formula (55″), instead of \(i_2\), one will have to substitute \(i_{\mathrm{vn}}\), the heat content of water taken from an outside source, and, of course,

\[ i_{\mathrm{vn}}<i_2. \]

III. Condensing plant with regeneration, producing only electrical energy

In this case

\[ \Sigma Q_{\mathrm{pol}}=860N_e; \]

\[ \begin{aligned} \Sigma Q_{\mathrm{pot}}={}& D_k^p(1-\Sigma\alpha_{mo})(i_k-i_2') +r_1D_k^p(1-\Sigma\alpha_{mo})(i_0-i_2') \\ &+r_1D_k^p\Sigma\alpha_{mo}(i_0-i_{mo}) \\ &+(1-\eta_m\eta_g)D_k^p\bigl[(i_0-i_k)-\Sigma\alpha_{mo}(i_{mo}-i_k)\bigr]; \end{aligned} \]

\[ \eta_t=\frac{\Sigma Q_{\mathrm{pol}}}{\Sigma Q_{\mathrm{pol}}+\Sigma Q_{\mathrm{pot}}}. \]

Taking into account the equalities:

\[ 860N_e=D_k^p\bigl[(i_0-i_k)-\Sigma\alpha_{mo}(i_{mo}-i_k)\bigr]\eta_m\eta_g; \]

\[ D_k^p(1-\Sigma\alpha_{mo})(i_k-i_2') +D_k^p\bigl[(i_0-i_k)-\Sigma\alpha_{mo}(i_{mo}-i_k)\bigr] = \]

\[ =D_k^p\bigl[(i_0-i_2')-\Sigma\alpha_{mo}(i_{mo}-i_2')\bigr]; \]

\[ r_1D_k^p(1-\Sigma\alpha_{mo})(i_0-i_2') +r_1D_k^p\Sigma\alpha_{mo}(i_0-i_{mo}) = \]

\[ =r_1D_k^p\bigl[(i_0-i_2')-\Sigma\alpha_{mo}(i_{mo}-i_2')\bigr]; \]

we may write for \(\eta_t\)

\[ \eta_t= \frac{860N_e} {D_k^p\bigl[(i_0-i_2')-\Sigma\alpha_{mo}(i_{mo}-i_2')\bigr](1+r_2)}. \tag{56} \]

In the absence of regeneration

\[ D_k^p=D_k;\qquad \Sigma\alpha_{mo}=0 \]

and formula (56) becomes the known formula (53).

From formula (56), however, one cannot draw the direct conclusion that regeneration leads to an increase in \(\eta_t\). Let us try to prove the latter.

For this purpose let us compare \(\eta_t\) for two turbine plants having identical initial and final parameters, as well as identical \(\eta_m\), \(\eta_g\), \(\eta_{oi}\).

Plant I. Operates without regeneration. Steam consumption \(D_k\).

Plant II. Operates with regeneration. Steam consumption \(D_{kp}\).

Let both plants cover the same electrical load of the district and have the same relative consumption of electrical energy for auxiliary needs—\(r_1\). The latter, of course, is a simplification that we make deliberately. It would also have been possible not to make this simplification.

For the turbine without regeneration

\[ N_e^k=\frac{D_k(i_0-i_k)}{860}\eta_m\eta_g. \]

For a turbine with regeneration,

\[ N_{\mathrm{e}}^{\mathrm{r}} = \frac{D_{\mathrm{k}}^{\mathrm{r}}\left[(i_0-i_{\mathrm{k}})-\sum \alpha_{m0}(i_{m0}-i_{\mathrm{k}})\right]}{860}\eta_{\mathrm{m}}\eta_{\mathrm{t}}; \]

\[ N_{\mathrm{e}}^{\mathrm{k}}=N_{\mathrm{e}}^{\mathrm{r}}=N_{\mathrm{e}}; \]

therefore

\[ D_{\mathrm{k}}(i_0-i_{\mathrm{k}})=D_{\mathrm{k}}^{\mathrm{r}}\left[(i_0-i_{\mathrm{k}})-\sum \alpha_{m0}(i_{m0}-i_{\mathrm{k}})\right], \]

whence

\[ D_{\mathrm{k}}^{\mathrm{r}}=D_{\mathrm{k}}\, \frac{i_0-i_{\mathrm{k}}}{(i_0-i_{\mathrm{k}})-\sum \alpha_{m0}(i_{m0}-i_{\mathrm{k}})} . \tag{57} \]

For a turbine without regeneration,

\[ \eta_{\mathrm{t}}^{\mathrm{k}}= \frac{860N_{\mathrm{e}}}{D_{\mathrm{k}}(i_0-i_2')(1+r_1)} \tag{53} \]

For a turbine with regeneration,

\[ \eta_{\mathrm{t}}^{\mathrm{r}}= \frac{860N_{\mathrm{e}}} {D_{\mathrm{k}}^{\mathrm{r}}\left[(i_0-i_2')-\sum \alpha_{m0}(i_{m0}-i_2')\right](1+r_1)} \tag{56} \]

Divide expression (53) by (56) term by term:

\[ \frac{\eta_{\mathrm{t}}^{\mathrm{k}}}{\eta_{\mathrm{t}}^{\mathrm{r}}} = \frac{D_{\mathrm{k}}^{\mathrm{r}}\left[(i_0-i_2')-\sum \alpha_{m0}(i_{m0}-i_2')\right]} {D_{\mathrm{k}}(i_0-i_2')}. \]

Substituting the value of \(D_{\mathrm{k}}^{\mathrm{r}}\) from (57), we find:

\[ \frac{\eta_{\mathrm{t}}^{\mathrm{k}}}{\eta_{\mathrm{t}}^{\mathrm{r}}} = \frac{(i_0-i_{\mathrm{k}})\left[(i_0-\sum \alpha_{m0}(i_{m0}-i_2')\right]} {(i_0-i_2')\left[(i_0-i_{\mathrm{k}})-\sum \alpha_{m0}(i_{m0}-i_{\mathrm{k}})\right]} \]

\[ \eta_{\mathrm{t}}^{\mathrm{k}} = \eta_{\mathrm{t}}^{\mathrm{r}} \frac{(i_0-i_{\mathrm{k}})\left[(i_0-i_2')-\alpha_{m0}(i_{m0}-i_2')\right]} {(i_0-i_2')\left[(i_0-i_{\mathrm{k}})-\sum \alpha_{m0}(i_{m0}-i_{\mathrm{k}})\right]}; \tag{58} \]

\[ \eta_{\mathrm{t}}^{\mathrm{k}} = \eta_{\mathrm{t}}^{\mathrm{r}} \frac{ 1-\dfrac{\sum \alpha_{m0}(i_{m0}-i_2')}{i_0-i_2'} }{ 1-\dfrac{\sum \alpha_{m0}(i_{m0}-i_{\mathrm{k}})}{i_0-i_{\mathrm{k}}} }. \tag{59} \]

From expression (59) one can establish the limiting cases in which

\[ \eta_{\mathrm{t}}^{\mathrm{k}}=\eta_{\mathrm{t}}^{\mathrm{r}}. \]

1) in the absence of steam extractions for regeneration,

\[ \sum \alpha_{m0}=0 \]

and

\[ \eta_{\mathrm{t}}^{\mathrm{k}}=\eta_{\mathrm{t}}^{\mathrm{r}}. \]

2) if there is heating by steam branched off before the turbine, then again no advantages are obtained.

\[ \sum \alpha_{m0}=\alpha, \]

\[ i_{m0}=i_0, \]

\[ \eta_{\mathrm{t}}^{\mathrm{k}}=\eta_{\mathrm{t}}^{\mathrm{r}}\frac{1-\alpha}{1-\alpha}=\eta_{\mathrm{t}}^{\mathrm{r}}. \]

The form of the widely known Noack curves \([16, 14]\) is explained by means of equation (59). (Limiting case 2).

If, for example, we wish to heat water in a mixing heater, using one point of steam extraction, and set ourselves the task of obtaining a water-heating temperature up to the saturation temperature corresponding to the steam pressure in the boiler, then, obviously, the steam for this purpose will have to be extracted by branching it off from the steam line, i.e., without passing it through the turbine. As has been shown, there will be no thermotechnical advantage from such heating.

It can be shown that, with the exception of the two limiting cases considered, regeneration leads to an increase in \(\eta_t\).

Fig. 17

In order that there be

\[ \eta_t^{p} > \eta_t^{k} \]

the following condition is necessary:

\[ 1 - \frac{\sum \alpha_{mo}(i_{mo} - i_2')}{i_0 - i_2'} < 1 - \frac{\sum \alpha_{mo}(i_{mo} - i_k)}{i_0 - i_k} \]

or

\[ \frac{\sum \alpha_{mo}(i_{mo} - i_2')}{i_0 - i_2'} > \frac{\sum \alpha_{mo}(i_{mo} - i_k)}{i_0 - i_k}; \]

multiplying both sides of the last inequality by

\[ \frac{i_0 - i_k}{\sum \alpha_{mo}(i_{mo} - i_2')} \]

as by a quantity that is certainly positive, we find

\[ \frac{i_0 - i_k}{i_0 - i_2'} > \frac{\sum \alpha_{mo}(i_{mo} - i_k)} {\sum \alpha_{mo}(i_{mo} - i_2')}. \tag{60} \]

The last inequality always holds by virtue of the second law of thermodynamics, since any of the values \(i_{mo}\) is less than \(i_0\). This inequality holds even in the case when one of the values \(i_{mo}\) is equal to \(i_0\), with the number of extractions for regeneration greater than one (Fig. 17)\(^{1}\).

This again explains the nature of the course of the Hoak curves constructed for a number of extractions for regeneration greater than unity. The curves have only one zero point, corresponding to the absence of extractions for regenerative heating of the water. When the water is heated to the saturation temperature successively in several heaters supplied with steam from several extraction points, we shall always have a gain from regeneration, although the last heater in the direction of water flow will be supplied with live boiler steam.

It is clear that this last heater will not give any benefit from the thermotechnical point of view, and its use should not be adopted.

\(^{1}\) The existence of inequality (60) can also be proved formally.

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