On Discrepancies in Determining the Efficiency of CHP Plants
Abstract Generated abstract
The article examines disputes over how to determine the efficiency of combined heat and power plants, contrasting a general heat-balance approach with systems that separate electric and heat indicators or rely on purely thermodynamic measures. It derives and defends an averaged efficiency expression based on the plant heat balance, arguing that losses in turbines, heat transport, boilers, and internal thermal schemes must be allocated to both electricity and heat supply rather than excluded from heat output. The paper criticizes alternative views associated with Gorshkov, Gokhshtein, and Fuks, contending that their methods either obscure the integral process of the CHP plant or introduce impractical conventions, while the proposed formulation provides a definite operational and economic indicator linked to fuel consumption.
Full Text
On Discrepancies in Determining the Efficiency of CHP Plants
I. N. Butakov
If, with respect to the determination of efficiency for a condensing power station,
\[ \eta_k=\frac{860\,E}{Q_{\mathrm{fuel}}} \]
there are no disagreements among power engineers, then when the question concerns the efficiency of a CHP plant, there are at present major disagreements.
It would seem natural, for a CHP plant, to proceed from its heat balance, which in the simplest case of a CHP plant consisting of a condensing part and a back-pressure part (or a part with worsened vacuum), with full utilization of the heat of the exhaust steam, may be written as follows:
\[ Q_{\mathrm{fuel}}=Q_k+Q_t, \tag{1} \]
where \(Q_k\) is the heat expended on the condensing part, \(Q_t\) the same for the back-pressure part, and \(Q_{\mathrm{fuel}}\) the total heat expenditure at the CHP plant over the time interval under consideration, for example, over a year.
Equation (1) is also represented in another form, namely:
\[ \frac{860E+Q_{\mathrm{out}}}{\eta} = \frac{860(E-E_t)}{\eta_k} + \frac{860E_t+Q_{\mathrm{out}}}{\eta_t}, \tag{2} \]
where \(E\) is the generation of kWh over the period under consideration by the whole CHP plant, \(E_t\) the same only for heat consumption, \(Q_{\mathrm{out}}\) the heat output at the station for heat consumers, and \(\eta_k\), \(\eta_t\), and \(\eta\) are the efficiencies: of the condensing part; of the back-pressure part with full utilization of heat; and \(\eta\) is the averaging efficiency of the whole CHP plant, to be determined.
For the values of the efficiencies \(\eta_k\) and \(\eta_t\) we have:
\[ \eta_k=\eta_{ku}\cdot\eta_{\mathrm{fuel}}\cdot\eta_t\cdot\eta_{oi}\cdot\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}}, \tag{3} \]
\[ \eta_t = \eta_{ku}\cdot\eta_{\mathrm{fuel}}\, \frac{(i_1-i_2)\eta_{oi}\cdot\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}} +i_1-(i_1-i_2)\eta_{oi}-t_{\mathrm{cond}}}{i_1-t_{\mathrm{cond}}} = \]
\[ = \left[ 1-\frac{(i_1-i_2)\eta_{oi}}{i_1-t_{\mathrm{cond}}} \left(1-\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}}\right) \right] \cdot \eta_{ku}\cdot\eta_{\mathrm{fuel}} = \eta_{ku}\cdot\eta_{\mathrm{fuel}}\cdot\eta'_{\mathrm{mech}}\cdot\eta'_{\mathrm{gen}}. \tag{4} \]
By \(t_{\mathrm{cond}}\) is meant, of course, the temperature of the condensate in the boilers. In this case, of course, \(\eta'_{\mathrm{mech}}\cdot\eta'_{\mathrm{gen}}>\eta_{\mathrm{mech}}\cdot\eta_{\mathrm{gen}}\). Here, as we shall see below, besides \(\eta'_{\mathrm{mech}}\cdot\eta'_{\mathrm{gen}}\), the efficiency \(\eta_{\mathrm{fuel}}\) is also debatable—the efficiency of the heat flow during its transport within the CHP plant, characterizing the heat losses to the surrounding environment from steam lines, the turbine casing, the feed-water circuit, including the regenerative system with its heaters, pipelines, the boiler and steam-conversion systems with their pipelines and pumps, characterizing, in a word, the losses of the entire heat scheme of the CHP plant.
The ideologist of the Ministry of Power Stations in this field, A. S. Gorshkov, in his book Technical and Economic Indicators of Thermal Electric Stations (1949, pp. 55–56), considers that no losses should be charged to the heat supplied ...
losses are not included, so that the efficiency of the turbogenerator with respect to heat output is \(\eta'_m = 1\). This is also justified by saying that heat is produced not in the turbine but in the boiler. From this point of view it follows that changes in the initial steam parameters, and deterioration of the internal relative efficiency of the turbine in the back-pressure section, do not affect the efficiency \(\eta'_m\) with respect to heat output. What is completely overlooked here is that the turbine acts here as a reducer, lowering the steam pressure from the boiler pressure to the pressure at the extraction point; consequently the losses of the reducer, i.e., in this case the mechanical and electrical losses of the turbogenerator, must inevitably be reflected in the heat consumers, just as its relative efficiency and the initial steam parameters are, as is indicated by formula (4).
“Heat energy must be characterized,” writes A. S. Gorshkov (p. 56), “...by the techno-economic indicators of the boiler house and boiler department, and not of the turbine shop”; yet again it is ignored that, on the one hand, the condensate of the boilers, through the thermal scheme and the boiler, exerts its influence on electric energy as well, while, on the other hand, the losses of the entire thermal scheme, again through the boiler, also affect the heat consumers. Therefore the losses of the entire thermal scheme must be evaluated by the overall efficiency \(\eta_{pot}\) of the heat flow during its transport within the CHP plant, and, consequently, the efficiency
\[ \eta_m = \eta_{ky}\cdot \eta_{pot}\cdot \eta'_{mech}\cdot \eta'_{gen} \]
(formula 4), and not
\[ \eta_m = \eta_{ky}\cdot \eta_{boil.}, \]
as in A. S. Gorshkov.
Acting in the role of an agnostic, A. S. Gorshkov denies the possibility of obtaining a general indicator for evaluating the CHP plant as a whole, since all attempts, in A. S. Gorshkov’s opinion, have a common fundamental error due to confusion of the concepts of the plant’s productivity and economy; A. S. Gorshkov illustrates this with the example of the “equivalence method” (Prof. Grinevetsky’s method). The shortcomings of Prof. Grinevetsky’s method have long been known, and this method, taken in isolation, is not useful for operational analysis, as A. S. Gorshkov himself says on p. 62 of his book. But the author’s argument on p. 61, supported by reference to the formula of the “physical” method advertised by A. S. Gorshkov,
\[ q_a^{f}=\frac{Q^{e}}{860\cdot W} =\frac{Q_n-Q_k-Q^m}{860\cdot W} =\frac{860\cdot W+q_c}{860\cdot W} =1+\frac{q_c}{860\cdot W}, \]
is undoubtedly flawed, because the quantity \(q_c\), characterizing losses in the condenser with the cooling water (p. 25), will vary depending on the value of \(Q_m\), so that \(q_c/860\cdot W\) will also be a variable quantity, as will the ratio
\[ \frac{Q_m}{860\cdot W}. \]
In the notation of formula (1), the latter quantity will be
\[ \frac{Q_{out}}{860\cdot E}. \]
This ratio is not at all a productivity factor, but a dimensionless energy coefficient whose magnitude, as all power engineers know, substantially affects the efficiency of a CHP plant.
Having rejected Prof. Grinevetsky’s formula for the efficiency of a CHP plant, which in our notation has the form
\[ \eta=\frac{860E+Q_{out}}{Q_{fuel}} =\frac{1+Q_{out}/860E}{Q_{fuel}/860E}, \]
A. S. Gorshkov became a defender and propagandist of a dualistic system of isolated indicators for a CHP plant: the efficiency of electric-energy generation and the efficiency with respect to heat output. The possibility of obtaining an overall efficiency...
the CHP plant for evaluating the process as a whole, as indicated above, is denied by A. S. Gorshkov, despite the direct indication of materialist dialectics that “the particular does not exist otherwise than in the connection which leads to the general” (Lenin, vol. XIII, p. 302). Meanwhile, the problem of finding a general indicator for a CHP plant is solved quite simply if one proceeds from its heat balance (1).
From equation (1) we have:
\[ \frac{1}{\eta} = \frac{1}{\eta_m} \left[ 1+ \frac{ \left(1-\frac{\mathcal{E}_m}{\mathcal{E}}\right) \left(\frac{\eta_m}{\eta_k}-1\right) }{ 1+\frac{Q_{\mathrm{vyr}}}{860\,\mathcal{E}} } \right]. \tag{5} \]
According to this formula, it is not the efficiency of the CHP plant that is determined, but its reciprocal, \(1/\eta\); not only because this gives a simpler expression, but also because the quantity \(1/\eta\), being directly proportional to fuel consumption, represents the heat expenditure in calories per 1 calorie used for the production of electric power and heat for heat consumers. Formula (2) is a combination of Prof. Grinevetsky’s method with the dualistic system of isolated indicators defended by A. S. Gorshkov, since
\[ Q_{\mathrm{topl}} = Q_k+Q_t = \left[ \frac{860(\mathcal{E}-\mathcal{E}_m)}{\eta_k} + \frac{860\,\mathcal{E}_m}{\eta_t} \right] + \frac{Q_{\mathrm{vyr}}}{\eta_t} = \frac{860\mathcal{E}+Q_{\mathrm{vyr}}}{\eta}, \]
i.e., here we have techno-economic indicators in two terms: separately for electric energy, in square brackets, and separately for thermal energy, with both of these terms connected by their common quantity
\[ \frac{860\mathcal{E}+Q_{\mathrm{vyr}}}{\eta}, \]
as their sum.
Thus A. S. Gorshkov has only outwardly, as it were, emancipated himself from the general averaging efficiency \(\eta\) of the CHP plant, but for this emancipation he paid the high price of losing an understanding of the integral process at the CHP plant, which led to an inflation of one side of the whole and directed him along the path of agnosticism.
Prof. D. P. Gokhshtein (Vestnik inzhenerov i tekhnikov, 1948, No. 4, p. 124) criticizes Prof. Grinevetsky’s formula
\[ \eta=\frac{860\mathcal{E}+Q_{\mathrm{vyr}}}{Q_{\mathrm{topl}}}, \]
attributing it for some reason to I. N. Butakov and V. T. Yurinsky, on the grounds that in its numerator completely incomparable terms (because of their heterogeneity) are summed—exhaust heat and mechanical work. Meanwhile, the “basement” part of the diagram in \(TS\) coordinates (the Rankine diagram) can be regarded not only as heat produced at the CHP plant for heat consumption, but also as the work performed by 1 kg of steam having initial temperature \(T_2\) and final temperature \(T'_2=0\), i.e., the quantities \(860\mathcal{E}\) and \(Q_{\mathrm{vyr}}\) may be regarded as homogeneous; and thus this objection of Prof. Gokhshtein must be understood as a well-known logical error bearing the name “use of a false premise” (error fundamentalis).
Thus, while criticizing others for “anti-thermodynamic” conceptions, Prof. Gokhshtein himself fell into the same sin, recommending for use the thermal efficiencies of power installations with utilization of exhaust heat (ibid., p. 129). In fact, if one takes, for example, his formula (5)
for a back-pressure engine using exhaust heat, it has the form
\[ \eta_{ym}=\frac{860+q_T\eta_T}{860/\eta_n}, \]
where \(q_T\) is the specific heat consumption for technological needs (per \(1\ \mathrm{kWh}\));
\(\eta_T\) is the thermal efficiency of the technological cycle;
\(\eta_n\) is the thermal efficiency of the back-pressure power cycle.
Thus, in the numerator, quantities are added which, in other notations, are again, as it were, incommensurable terms, just as in the formula of Prof. V. I. Grinevetskii, where \(Q_{\mathrm{out}}\) is understood to mean the heat intended for use by heat consumers, produced at the station, less the heat of the return water (mainly the latent heat of vaporization), and inevitably \(Q_{\mathrm{out}}/\mathcal{E}_m=q_T\eta_T\).
The proposal to interpret the degree of perfection of a CHP plant from the standpoint of thermodynamics encounters a number of difficulties. For example, formula (45) for the degree of perfection of Prof. G. I. Fuchs has the form:
\[ \mu=\frac{860N+\sum Q_r(1-T_0/T_{cm})}{m\cdot B\cdot Q_p} \]
(Izvestiya TPI, vol. 66, no. 2, p. 112).
Here \(T_0\) is the absolute temperature of the environment. The choice of the value \(T_0\) is difficult: CHP plants operate at different temperatures. Some conventional “normal” ambient temperature must be chosen. Prof. G. I. Fuchs, after analyzing average annual air temperatures for the USSR, proposes, for example, to take \(T_0=273\ K\), but at the same time emphasizes that \(T_0=273\ K\) may be adopted only when it is necessary to compare installations operating under different conditions. In tests, however, \(T_0\) should not be adopted, but measured as it is. The presence in the USSR of sharp temperature fluctuations depending on place and time of year increases the difficulty of estimating \(T_0\). Or, if we take the example of a CHP plant having the same technological, heat, and electric loads all year round, then according to formula (45) of Prof. G. I. Fuchs it turns out that in summer, at a temperature \(T_0=293\ K\), and in winter, at \(T_0=253\ K\), the degrees of perfection of the CHP plant will be different, whereas according to formula (5) the actual fuel consumption will be the same; i.e., the efficiency will remain unchanged.
The proponents of introducing purely thermodynamic evaluations into practice, and among them especially Prof. D. P. Gokhshtein, an enthusiast of the second law of thermodynamics, forget the essential point: thermodynamics is a branch of physics and is not yet engineering, which is a synthesis of physics and economics, treating a technical phenomenon with its special regularities. This synthesis, at every stage of production development, gives rise to its own designs of boiler units, determines the permissible diameter of steam lines, the number of regenerative heaters and their surfaces, the heating surfaces of boilers, the degree of cooling and temperature differences in condensers, and also their surfaces, the specific weights in \(\mathrm{kg}/\mathrm{kW}\) of turbine units, etc.
Therefore, the use of equality (2) and formula (5) is entirely legitimate, all the more so since the latter is capable of reflecting thermodynamic tendencies toward reducing the increase of entropy in boiler units, in boilers, in steam lines, condensers, etc., because a change in entropy exerts its influence on the generation of electric power in heat consumption \(\mathcal{E}_m\), on the efficiencies \(\eta_k\) and \(\eta_m\).
Since in both parts of equation (2) there are quantities proportional to fuel consumption, an economic interpretation of this equation is possible
...of equation (2) of the heat balance, since the fuel consumption, for its part, is proportional to the expenditure of socially necessary labor for its extraction. For this reason the averaging efficiency may characterize also the degree of utilization of socially necessary labor with respect to the principal fuel component of the monetary expenditures of a CHP plant, and from this point of view may be called the economic efficiency of a CHP plant.
Criticizing formula (5), Prof. G. I. Fuks (Izvestiya TPI, vol. 66, issue 2, p. 91) indicates that the very values \(\eta_k\) and \(\eta_m\) for a given thermal installation are conditional quantities. Such a conclusion by the author of the article is incomprehensible. In fact, the efficiency \(\eta_m\) for a given installation is calculated by the quite definite formula (4). This makes it possible to obtain the fuel consumption for heat consumers in the form
\[ B_m=\frac{Q_{\mathrm{vyr}}}{\eta_k\cdot Q_r^n\cdot 10^3}\, t, \]
so that, with the known working consumption \(B_m\) at the CHP plant, the fuel consumption for electric consumers is determined in \(t\), as \(B_e=B-B_m\).
Hence
\[ \eta_e=\frac{860\cdot E}{B_e\cdot Q_r^n 10^3} \]
is the overall efficiency for the generation of electric power at the CHP plant.
To determine now the efficiency \(\eta_k\) of the condensing part of the CHP plant in operation, it is necessary to know the fuel consumption for this part, which is equal to
\[ B_k=B-B_m-\frac{860\cdot Q_{\mathrm{vyr}}}{10^3\cdot q\cdot \eta_m\cdot Q_r^n}\, t, \]
where
\[ B_m^e=\frac{860\cdot Q_{\mathrm{vyr}}}{10^3\cdot q\cdot \eta_m\cdot Q_r^n} \]
is the consumption in \(t\) of fuel for the generation of electric energy \(E_m\) kWh in heat consumption, \(q=\frac{Q_{\mathrm{vyr}}}{E_m}\) is the number of calories obtained from each kWh in heat consumption for the needs of the latter. Then the efficiency of the condensing part is
\[ \eta_k=\frac{860(E-E_m)}{B_k\cdot Q_r^n\cdot 10^3} = \frac{860\left(E-\frac{Q_{\mathrm{vyr}}}{q}\right)}{B_k\cdot Q_r^n\cdot 10^3}, \]
i.e., a quantity not conditional, but quite definite for the given CHP plant.
Equally incomprehensible is the second assertion of Prof. G. I. Fuks that, “in order to obtain the corresponding result, the author had to introduce the assumption that different variants of the installation are compared with one another at the same generation of electric power, the deficiency of the latter being covered by an additional condensing installation with efficiency \(\eta_k\).” In comparing variants, the quantity specified for the CHP plant is not the generation of electric power, but the energy coefficient \(Q_{\mathrm{vyr}}/860E\), which quantity is included in formula (5) and in other formulas that include independent boiler houses as well as current from hydroelectric stations. Problems for CHP plants and for the power system must necessarily be solved taking precisely the energy coefficient as the initial datum.
Why Prof. G. I. Fuks considers these conditions, indicated above, to be “arbitrary,” preventing the obtaining of substantiated proof of the correctness of formula (5), which for some reason the author has confused with the formula of Prof. Grinevetsky, writing it in his own notation as
\[ \eta=\frac{860\cdot N_e+\Sigma Q_4}{B\cdot Q_r^n}, \]
is completely unclear.
Here he applied the well-known paralogism of “substitution of proof” (ignoratio elenchi), which was widely used in his time by A. S. Gorshkov, who, in No. 6 for 1948 of the journal Electric Stations, came forward with criticism of my allegedly own formulations; moreover, he constructed his objections while polemicizing also against the formulas of Prof. Grinevetsky.
Thus his objections missed the mark, and the editors of the departmental journal refused to publish my protest against the use of sophistical devices in scientific polemics.
Comrade Stalin considered it necessary to emphasize sharply this formula, drawn from centuries of experience: “that no science can develop and prosper without a struggle of opinions, without freedom of criticism. But this generally recognized rule was ignored and trampled upon in the most unceremonious manner.”
It was ignored and trampled upon because there existed an “Arakcheev regime,” which cultivated irresponsibility and encouraged such outrages in some editorial offices of our technical journals.