ON THE ISSUE OF DETERMINING THE FUEL COMPONENT FOR HEAT CONSUMERS AT A CHP PLANT

I. N. Butakov

In the 1955 Ministry of Power Plants instruction for compiling the technical report on the operation of thermal power stations, for determining the consumption, in tons of standard fuel, at a CHP plant for the supply of heat to outside consumers, the following formula is given (Instruction, item 3, p. 10):

\[ B_{\mathrm{t.e.}}= \frac{\left(\sum Q_{\mathrm{otp}}^{\mathrm{p}}+\sum Q_{\mathrm{otp}}^{\mathrm{gv}}-\sum Q_{\mathrm{voz}}+\sum Q_{\mathrm{pot}}\right)\cdot 10^3} {7000\cdot \eta_k^{\mathrm{t}}}. \tag{1} \]

Here \(\sum Q_{\mathrm{otp}}^{\mathrm{p}}\) is the total supply of heat in the form of steam of various parameters to outside consumers, both directly from the boiler house and from extractions or back pressure of turbines, in thousand kcal; \(\sum Q_{\mathrm{otp}}^{\mathrm{gv}}\) is the same with hot water, regardless of where the steam went for the water heaters, in million kcal; \(\sum Q_{\mathrm{voz}}\) is the return of heat from consumers with condensate, in million kcal; \(\sum Q_{\mathrm{pot}}\) is heat losses in million kcal in connection with the supply of heat to outside consumers (item 3 of the Instruction); \(\eta_k^{\mathrm{t}}\) is the net efficiency of the thermal boiler house (item 56 of the Instruction).

In formula (1) there is a fundamentally incorrect equalization made with respect to heat consumers, which have different efficiencies depending on whether they are supplied from turbine extractions or directly from boilers, and, moreover, there is an incomplete accounting of the losses of the entire heat scheme of the CHP plant, since \(\sum Q_{\mathrm{pot}}\) accounts for them only partially. As a result, the actual fuel consumption for heat consumers will be somewhat greater than that determined by formula (1).

In his article (Electric Stations, 1957, No. 12), L. M. Mittelman introduced a correction into formula (1), adding a second term consisting of the sum of fuel expenditures for electric power for in-house needs attributable to heat consumers, namely:

\[ \left(E_{\mathrm{set}}^{\mathrm{nas}}+E_{\mathrm{pod}}^{\mathrm{nas}}+E_{\mathrm{vn}}^{\mathrm{nas}}\right)\cdot 10^{-4}\cdot b_e^{\mathrm{ot}}, \]

where \(E_{\mathrm{set}}^{\mathrm{nas}}\) is the consumption of electric power, in kWh, for network pumps; \(E_{\mathrm{pod}}^{\mathrm{nas}}\) is the same for feed pumps; \(E_{\mathrm{vn}}^{\mathrm{nas}}\) is the same for condensate pumps of network water heaters, and \(b_e^{\mathrm{ot}}\) is the specific consumption, in kg/kWh, of standard fuel for the released electric power. Thus, the amend-

as P. M. Mittelman gives the fuel consumption taking into account the consumption of electric power for auxiliary needs, thereby placing heat consumers in an even more difficult position. But such a correction corresponds to reality and should not give rise to objections.

In the preceding period, many objections were raised against the existing method of distributing fuel at a CHP plant between electric and heat consumers. Thus, for example, it was pointed out that this method assigns the benefit from the combined production of heat and electric power, with respect to fuel consumption, to the electric subscribers, greatly reducing the cost of a kWh and placing heat consumers in a difficult position. In connection with this it was further noted that this could even hinder the spread of CHP plants. It was also pointed out that the cost of a megacalorie here does not depend on the steam parameters, since the latent heat of the latter, utilized by heat subscribers, depends little on the steam pressure in the extractions. This, it was said, contradicts common sense.

To understand all these questions better, let us write the heat balance equation of a CHP plant, taking some interval of time, for example a year:

\[ \frac{860 \cdot E_{t} + Q_{\mathrm{otb}}}{\eta_{t}} + \frac{860\,(E - E_{t})}{\eta_{k}} + \frac{Q_{\mathrm{kot}}}{\eta_{ku}\cdot \eta_{\mathrm{pot}}} = \frac{860 \cdot E + Q_{\mathrm{otb}} + Q_{\mathrm{kot}}}{\eta}. \tag{2} \]

Here \(E\) is the total generation of electric power, kWh; \(E_{t}\) is the same for heat consumption \(Q_{\mathrm{otb}}\) from turbine extractions, kcal; \(Q_{\mathrm{kot}}\) is the heat supplied directly from boilers, kcal; \(\eta_{k}\) is the efficiency of the condensing part, \(\eta_{t}\) of the back-pressure part, \(\eta_{ku}\) of the boiler installation, \(\eta_{\mathrm{pot}}\) of the heat flow inside the CHP plant, and \(\eta\) the overall economic efficiency of the CHP plant.

Let us select the terms of the left-hand side of expression (2), by heat and fuel consumption, that depend on heat consumption, knowing that \(Q_{\mathrm{otb}} = q \cdot E_{t}\), where \(q\) is the number of kilocalories in the heat of exhaust steam per 1 kWh on the basis of heat consumption \(Q_{\mathrm{otb}}\). Then\(^1\)

\[ \frac{Q_{\mathrm{kot}}}{\eta_{ku}\cdot \eta_{\mathrm{pot}}} + \frac{Q_{\mathrm{otb}}}{\eta_{t}} \left[ 1 - \frac{860}{q} \left( \frac{\eta_{t}}{\eta_{k}} - 1 \right) \right] = Q_{Q}\ \frac{\mathrm{kcal}}{\mathrm{year}}. \]

\(^1\) The physical meaning of the second term of the expression for \(Q_{Q}\) is that the generation of electric power for heat consumption \(E_{t}\) kWh displaces the same number of condensing kWh, giving over the year a fuel heat saving of

\[ 860 \cdot E_{t}\left(\frac{1}{\eta_{k}} - \frac{1}{\eta_{t}}\right) = \frac{860 \cdot Q_{\mathrm{otb}}}{q} \left(\frac{1}{\eta_{k}} - \frac{1}{\eta_{t}}\right) \ \mathrm{kcal}. \]

Thus, the heat consumers receiving steam from turbine extractions will be charged, over the year, not with the heat consumption in fuel \(Q_{\mathrm{otb}}/\eta_{t}\), but with

\[ \frac{Q_{\mathrm{otb}}}{\eta_{t}} - \frac{860 \cdot Q_{\mathrm{otb}}}{q} \left(\frac{1}{\eta_{k}} - \frac{1}{\eta_{t}}\right) = \frac{Q_{\mathrm{otb}}}{\eta_{t}} \left[ 1 - \frac{860}{q} \left( \frac{\eta_{t}}{\eta_{k}} - 1 \right) \right], \]

i.e., the total consumption of fuel heat falling on heat subscribers will be

\[ Q_{Q} = \frac{Q_{\mathrm{kot}}}{\eta_{ku}\cdot \eta_{\mathrm{pot}}} + \frac{Q_{\mathrm{otb}}}{\eta_{t}} \left[ 1 - \frac{860}{q} \left( \frac{\eta_{t}}{\eta_{k}} - 1 \right) \right]. \]

whence

\[ \frac{Q_{\mathrm{kot}}}{Q_{\mathrm{otb}}}\cdot \frac{1}{\eta_{\mathrm{ku}}\cdot\eta_{\mathrm{pot}}}\cdot \frac{1}{\eta_t} \left[ 1-\frac{860}{q}\left(\frac{\eta_t}{\eta_k}-1\right) \right] = \frac{Q_Q}{Q_{\mathrm{otb}}}. \tag{3} \]

The dependence of \(Q_Q/Q_{\mathrm{otb}}\) on \(q\) is hyperbolic; if \(q=\infty\), then the horizontal asymptote is

\[ Q_Q:Q_{\mathrm{otb}}= \frac{Q_{\mathrm{kot}}}{Q_{\mathrm{otb}}}\cdot \frac{1}{\eta_{\mathrm{ku}}\cdot\eta_{\mathrm{pot}}}\cdot \frac{1}{\eta_t}. \]

As can be seen, it depends on the ratio \(Q_{\mathrm{kot}}:Q_{\mathrm{otb}}\). If \(Q_Q/Q_{\mathrm{otb}}=\infty\), then the vertical asymptote is \(q=0\). The point for constructing the hyperbola is found at its intersection with the \(q\)-axis, when \(Q_Q/Q_{\mathrm{otb}}=0\), and, consequently,

\[ 1/\eta_t+ \frac{Q_{\mathrm{kot}}}{Q_{\mathrm{otb}}}\cdot \frac{1}{\eta_{\mathrm{ku}}\cdot\eta_{\mathrm{pot}}} = \frac{860}{\eta_t\cdot q}(\eta_t\eta_k-1), \]

whence

\[ q= \frac{860/\eta_t(\eta_t\eta_k-1)} {Q_{\mathrm{kot}}/Q_{\mathrm{otb}}\cdot 1/\eta_{\mathrm{ku}}\cdot\eta_{\mathrm{pot}}+1/\eta_t}. \]

Let \(\eta_t=0.8\); \(\eta_{\mathrm{ku}}\cdot\eta_{\mathrm{pot}}=0.82\). Then for

\[ \frac{Q_{\mathrm{kot}}}{Q_{\mathrm{otb}}}=0.5 \]

we obtain the asymptote \(Q_Q:Q_{\mathrm{otb}}=1.86\). For \(Q_{\mathrm{kot}}:Q_{\mathrm{otb}}=1\) the asymptote is

\[ \frac{Q_Q}{Q_{\mathrm{otb}}}=2.47. \]

If \(\eta_k=0.25\), then at

\[ \frac{Q_Q}{Q_{\mathrm{otb}}}=0 \]

and

\[ q=\frac{860/0.8(3.2-1)}{1.86\ \text{or}\ 2.47}=1270 \]

or 960.

In Fig. 1 these two hyperbolas are plotted for

\[ \frac{Q_{\mathrm{kot}}}{Q_{\mathrm{otb}}}=0.5 \]

and 1.0. But

\[ q=\frac{860}{H\cdot\eta_{\mathrm{oe}}}\,(i_1-\eta_{\mathrm{oi}}\cdot H\, t_{\mathrm{cond}}) \ \frac{\mathrm{kcal}}{\mathrm{kWh}}. \]

The expression in parentheses represents the latent heat of vaporization of the extraction steam, which changes little with changes in the pressure in the steam extractions. Thus, \(q\), depending on the heat drop up to the steam-extraction point, varies according to a law close to a hyperbola with asymptotes \(H=0\) and \(q=0\). Thus, raising the pressure of steam extraction, decreasing the heat drop \(H\cdot\eta_{\mathrm{oe}}\) up to the extraction point, entails an increase in \(q\), while an increase in \(q\) causes the value of \(Q_Q/Q_{\mathrm{otb}}\) in Fig. 1 to rise; moreover, for a given value of \(Q_{\mathrm{otb}}\), \(Q_Q\), i.e., the fuel consumption, inevitably increases.

It follows from what has been said that the heat consumption for heat consumers, at a specified heat demand, proves to be dependent on the pressure in the extraction, as well as on the initial parameters of the steam.

Fig. 1 gives the changes of \(Q_Q/Q_{\mathrm{otb}}\) with \(q\), but under practical conditions, as a rule, \(Q_Q\) must be referred not to \(Q_{\mathrm{otb}}\), but to \((Q_{\mathrm{otb}}+Q_{\mathrm{kot}})\), i.e., to the entire output of the CHP plant for heat subscribers, bearing in mind the necessity of cutting the peaks of the heat load, which, when \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}}=0.5\), gives the ratio

\[ \frac{Q_Q}{Q_{\mathrm{otb}}+Q_{\mathrm{kot}}} = \frac{Q_Q}{1.5\cdot Q_{\mathrm{otb}}}, \]

and at \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}} = 1.0\) this ratio becomes

\[ Q_Q = 2 \cdot Q_{\mathrm{otb}}. \]

Thus, in both of the indicated cases, this ratio of the consumption \(Q_Q\) of heat in the fuel for heat consumers to the heat \((Q_{\mathrm{otb}} + Q_{\mathrm{kot}})\) actually supplied to the latter as CHP-plant output proves to be smaller by the factor by which the value of \(q\) is smaller. In practice, the ratio \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}} = 0.5\) is encountered more often than the case

\[ \frac{Q_{\mathrm{kot}}}{Q_{\mathrm{otb}}} = 1.0. \]

To illustrate what has just been said, let us take a particular example in which \(q = 2875\ \mathrm{kcal/kWh}\), and consequently, at \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}} = 0.5\), in Fig. 1 we have \(Q_Q/Q_{\mathrm{otb}} = 1.0\), and

\[ \frac{Q_Q}{1.5 \cdot Q_{\mathrm{otb}}} = 1 : 1.5 = 0.66\ \mathrm{kcal/kcal}, \]

where

\[ Q_Q < Q_{\mathrm{kot}} \eta_{\mathrm{ku}} \eta_{\mathrm{tr}} + Q_{\mathrm{otb}} \eta_{\mathrm{tp}}. \]

Figure labels: vertical axis \(Q_e/Q_{\mathrm{otb}}\); horizontal axis \(q\). Marked levels: \(Q_2\), \(2.47\), \(2.0\), \(1.86\), \(1.0\), \(0\). Curve annotations: “Asymptote for \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}} = 1.0\)”; “Asymptote for \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}} = 0.5\)”; \(\eta_{\mathrm{kot}}/Q_{\mathrm{otb}} = 1.0\); \(\eta_{\mathrm{kot}}/Q_{\mathrm{otb}} = 0.5\). Horizontal-axis marks: \(1000\), \(2000\), \(3000\), \(4000\), \(5000\).

Fig. 1.

At \(Q_Q/Q_{\mathrm{otb}} < 1\), i.e., at \(q < 2875\ \mathrm{kcal/kWh}\), the specific heat consumption assigned to the entire CHP-plant output for heat consumers \((Q_{\mathrm{otb}} + Q_{\mathrm{kot}})\) will be less than \(0.66\ \mathrm{kcal/kcal}\), or less than \(0.66 \cdot 10^6/7000 = 94.5\ \mathrm{kg/Mcal}\) of standard fuel per Mcal; and, conversely, at \(Q_Q/Q_{\mathrm{otb}} > 1\), it increases relative to \(0.66\ \mathrm{kcal/kcal}\) and reaches the specific consumption value of \(170\ \mathrm{kg/Mcal}\), usually determined according to the current instruction, at \(Q_Q/Q_{\mathrm{otb}} = 170/94.5 = 1.8\). This corresponds in Fig. 1, when \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}} = 1.0\), to approximately \(q = 4000\ \mathrm{kcal/kWh}\); and when \(Q_{\mathrm{kot}}/Q_{\mathrm{otb}} = 0.5\), the value of \(q\) will be close to infinity, while the same specific consumption of \(170\ \mathrm{kg/Mcal}\) cannot in practice be reached at all. Thus, with the methodology under consideration, fuel consumption will be distributed more evenly among heat-

and electric consumers. Since \(\frac{Q_0}{Q_{\mathrm{otb}} + Q_{\mathrm{kot}}}\) is a fraction, and the heat subscribers will now bear a smaller share of the consumed fuel as the share of electric consumers increases, the objection concerning obstacles hindering the spread of CHP plants will be removed. Such large heat consumers as vegetable-processing combines, or work connected with melting snow in large cities, at factories and at railway stations, where the removal and carting away of snow is expensive, can receive heat at a significantly lower price. Moreover, the proposals that have appeared recently to replace CHP plants with boiler houses at the points of heat consumption, with electricity obtained from powerful power systems—that is, proposals to use separate installations—will disappear. The latter is due to the lack of distinction, under the current methodology of the MES, in obtaining heat for heat consumers both from turbine extractions and directly from boilers, when it is asserted that heat from turbine extractions is likewise obtained from boilers, and that turbogenerators have nothing to do with it (Gorshkov A. S., Technical and Economic Indicators of Thermal Power Plants, pp. 56, 63, 1949).