PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 125　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1964

DETERMINATION OF THE FINAL PRESSURE IN STEAM-TURBINE INSTALLATIONS WHEN DESIGNING THERMAL POWER STATIONS

E. N. SHADRIN

(Submitted by the Department of Thermal-Power Installations)

As is known, most modern turbines are designed and manufactured for a design final pressure of \(0.03—0.035\) ata and a cooling ratio of \(55—65\). Meanwhile, in reality turbines may operate under various conditions both with respect to fuel and to water supply. Therefore, under conditions of actual design of thermal power stations it becomes necessary to determine the final steam parameters in steam-turbine installations, for which the water-supply system must then be designed. Even if in the future turbines and condensers are manufactured for different final parameters, the standard series of these parameters will not cover the entire variety of local concrete conditions, and each time in designing a thermal power station it will be necessary to find the optimum final pressure. Below a method is given for solving the indicated question as applied to stations with once-through water supply.

We determine the optimum final pressure \(p_k^{\mathrm{opt}}\) from the condition of minimum total production costs for the installation \(\sum S_{\min}\), and in the total production costs we shall include only those components that depend on the final pressure \(p_k\). Such components in the present case are the costs of fuel used to generate \(\mathcal{E}_{\text{yr}}\) of the annual amount of electric energy in MWh, the costs of pumping cooling water for steam condensation in the condenser, depreciation and current-repair charges for the water conduits and the pumping station. In addition, in the total costs we shall include a certain percentage of the capital expenditures for the water conduits and the pumping station in accordance with the coefficient of efficiency of capital expenditures adopted at the design stage.

Thus,

\[ \sum S = S_{\mathrm{t}}^{\mathrm{e}} + S_{\mathrm{t}}^{\mathrm{per}} + S_{k}^{\mathrm{w.c.}} + S_{k}^{\mathrm{p.s.}} + \delta \left(K_{\mathrm{w}} + K_{\mathrm{p}}\right) = f(P_k). \tag{1′} \]

We proceed to determine the individual terms of equation (1′). The total specific fuel consumption for generating \(1\) MWh of electric energy can be determined by the formula

\[ b_{\mathrm{e}}=\frac{1000\varepsilon_{\mathrm{p}}(i_0-i_{\mathrm{fw}})} {H_0\eta_{0\mathrm{e}}\eta_{\mathrm{ku}}\eta_{\mathrm{fl}}\eta_{\mathrm{aux}}Q_{\mathrm{p}}^{\mathrm{n}}} \ \text{kg/MJ}. \tag{1} \]

Here \(d_{\mathrm{e}}=\dfrac{1}{H_0\eta_{0\mathrm{e}}}\ \text{kg/kJ}\) is the specific steam consumption for a once-through turbine;

\(i_0\) is the initial heat content of the steam, \(\text{kJ/kg}\);

\(i_{\mathrm{fw}}\) is the heat content of the feedwater, \(\text{kJ/kg}\);

\(\eta_{\mathrm{u}}, \eta_{\mathrm{fl}}\) are the efficiencies, respectively, of the boiler unit and the flow path;

\(Q_{\mathrm{p}}^{\mathrm{n}}\) is the lower heating value, \(\text{kJ/kg}\);

\(\eta_{\mathrm{aux}}\) is the plant service factor, accounting for fuel losses at the station; it is determined by the type of equipment and the standard of operation; it varies within the limits from 0.8 to 0.95;

\(\eta_{0\mathrm{e}}\) is the relative electrical efficiency;

\(H_0\) is the available heat drop in the turbine, \(\text{kJ/kg}\);

\(\varepsilon_{\mathrm{p}}\) is a coefficient allowing for the increase in steam consumption due to regenerative extractions.

Since, in the case under consideration, the turbine is specified, it may be assumed that \(i_0, i_{\mathrm{fw}}, \eta_{0\mathrm{e}}, \eta_{\mathrm{ku}}, \eta_{\mathrm{fl}}, \eta_{\mathrm{aux}}\) are constant\(^1\), independent of the terminal pressure \(p_k\). The value \(Q_{\mathrm{p}}^{\mathrm{n}}\) depends on the grade of fuel burned and is determined by specific local conditions. Consequently, \(b_{\mathrm{e}}=f(H_0)=f'(p_k)\). As \(p_k\) decreases, \(H_0\) will increase, while \(b_{\mathrm{e}}\) will decrease.

With the decrease in fuel consumption, the annual monetary expenditures on fuel \(S_{\mathrm{t}}^{\mathrm{e}}\) will decrease; these are determined as

\[ S_{\mathrm{t}}^{\mathrm{e}}= \frac{3600\varepsilon_{\mathrm{p}}(i_0-i_{\mathrm{fw}})AN_{\mathrm{e}}R_{\mathrm{t}}} {H_0\eta_{0\mathrm{e}}\eta_{\mathrm{ku}}\eta_{\mathrm{fl}}\eta_{\mathrm{aux}}Q_{\mathrm{p}}^{\mathrm{n}}} \ \text{rub/year}. \tag{2} \]

Here \(A\) is the utilization factor of installed capacity, \(\text{h/year}\). Under base-load operation one may take \(A=6500\text{–}7200\ \text{h/year}\); under peak-load operation \(A=4000\text{–}5000\ \text{h/year}\);

\(N_{\mathrm{e}}\) is the specified electric power, \(\text{kW}\);

\(R_{\mathrm{t}}\) is the cost of a ton of coal, rub.

If the installation has steam reheat, then in the numerator the expression in parentheses will have the form

\[ i_0-i_{\mathrm{fw}}+\sum(i_{\mathrm{p}}'-i_{\mathrm{p}}''), \]

where \(i_{\mathrm{p}}'\) is the heat content of the steam after reheat, \(\text{kJ/kg}\);

\(i_{\mathrm{p}}''\) is the heat content of the steam before reheat, \(\text{kJ/kg}\).

Thus, by assigning different values of \(p_k\), we determine, for these values, \(S_{\mathrm{t}}^{\mathrm{e}}=f(p_k)\).

As is evident from equation (2), with a decrease in \(p_k\), expenditures on fuel will decrease. But, on the other hand, one can note a number of components of annual costs that will increase as \(p_k\) decreases. Thus, the costs of pumping cooling water will increase. These costs may be determined as follows.

The quantity of circulating water for steam condensation is

\[ W=\frac{10^{-3}\alpha N_{\mathrm{e}}(i_k-t_k)\varepsilon_{\mathrm{p}}} {[(t_s-t_1)-(t_s-t_2)]H_0\eta_{0\mathrm{e}}} \ \text{m}^3/\text{s}. \tag{3} \]

In this formula

\(^1\) Strictly speaking, with a change in terminal pressure the efficiency \(\eta_{0i}\) will change; however, since \(p_k\) varies within comparatively narrow limits, the change in \(\eta_{0i}\) may be neglected in these calculations.

\(\dfrac{\alpha \varepsilon_p N_e}{H_0 \eta_{0e}}\) — the amount of steam entering the condenser, kg/s;

\(\alpha\) — coefficient accounting for the reduction in steam flow to the condenser due to its extraction for regeneration, as compared with the total amount entering the turbine;

\(t_s\) — temperature of the exhaust steam, \({}^{\circ}\mathrm{C}\);

\(t_1\) and \(t_2\) — temperature of the circulating water at the inlet to the condenser and at the outlet from the condenser, \({}^{\circ}\mathrm{C}\);

\(i_k - t_k\) — amount of heat transferred to the cooling water during condensation of 1 kg of steam, kcal/kg.

In formula (3), the difference \(i_k - t_k\) depends only slightly on the pressure in the condenser and in calculations may be taken as constant. And since the temperature \(t_s\) and the available temperature difference \(H_0\) depend only on \(p_k\), \(W\) is determined by the final pressure. The decisive value in the dependence \(W = f(p_k)\) is the term \(t_s - t_1\). The deeper the vacuum, the smaller \(t_s - t_1\), and the larger \(W\). At \(t_s = t_1 + (t_s - t_2)\), \(W = \infty\).

In formula (3), the difference \(t_s - t_2\) represents the temperature head in the condenser; its value will vary with changes in the amount of cooling water, since the water velocity in the condenser tubes will change and, consequently, so will the heat-transfer coefficient. However, this change is small and may be neglected, taking the difference \(t_s - t_2\) as constant and equal to the design temperature head.

Since, for designing water-supply systems, it is necessary to know the largest amount of water, it is expedient to construct, from equation (3), the dependence \(W = f(p_k)\).

The consumption of electric power for pumping circulating water, in MJ per year, is

\[ E_{\mathrm{cw}} = \frac{3.6 \cdot 10^{-3}\varphi W H_{\mathrm{cw}} A}{\eta_n}. \tag{4} \]

Here \(H_{\mathrm{cw}}\) is the head overcome by the circulating pumps, N/m\(^2\); \(\eta_n\) is the pump efficiency.

The coefficient \(\varphi\) accounts for the additional amount of water supplied for cooling the oil and air. One may take \(\varphi = 1.04\text{--}1.08\).

The total head \(H_{\mathrm{cw}}\) consists of the head \(H_k\) expended in overcoming the resistance of the condenser, \(H_{\mathrm{wd}}\) of the supply circulating water conduits, and also the head \(H_g\), caused by the difference in elevations between the condenser and the water level in the water-supply source.

The resistance of the condenser may be determined by the formula

\[ H_k = z \rho \left(0.031 \frac{L_k}{d_k}\frac{w_k^2}{2} + 1.4\frac{w_k^2}{2}\right) = z \psi w_k^2 \ \mathrm{N/m^2}, \tag{5} \]

where \(z\) is the number of condenser passes;

\(L_k\) is the length of the condenser tubes;

\(d_k\) is the internal diameter of the condenser tubes, m;

\(w_k\) is the water velocity in the condenser tubes, m/s;

In formula (5), the first term characterizes the resistance during the passage of water inside the condenser tubes, and the second term the local resistances in the condenser.

If the design value of the water velocity inside the condenser tubes is denoted by \(w_k^p\), then

\[ H_k = z\psi \left(\frac{w_k}{w_k^p}\right)^2 \cdot w_k^{p\,2}. \]

Since the flow cross-section of the condenser tubes \(f_{\mathrm{v}}\) is constant, then \(w \equiv W\) and

\[ H_{\mathrm{k}}=z\psi \left(\frac{W}{W_{\mathrm{p}}}\right)^2 w_{\mathrm{k}}^2 . \tag{6} \]

Thus, \(H_{\mathrm{k}}=f(W)=f'(p_{\mathrm{k}})\).

With deepening vacuum, when the value \(W\) increases, \(H_{\mathrm{k}}\) will increase.\(^1\)

The resistance of the water conduit is

\[ H_{\mathrm{vod}}=\lambda \frac{L_{\mathrm{e}}}{d_{\mathrm{v}}}\cdot \frac{w_{\mathrm{v}}^2\rho}{2}\ \mathrm{N/m^2}. \tag{7} \]

Here \(d_{\mathrm{v}}\) is the diameter of the water conduit; its value may be determined for a double water conduit by the formula

\[ d_{\mathrm{v}}=\frac{1.414\,W^{0.5}\varphi^{0.5}}{w_{\mathrm{v}}^{0.5}}\ \mathrm{m}. \tag{8} \]

For a single water conduit,

\[ d_{\mathrm{v}}=\frac{2W^{0.5}\varphi^{0.5}}{w_{\mathrm{v}}^{0.5}}\ \mathrm{m}. \tag{9} \]

Then, for a double water conduit, its resistance is

\[ H_{\mathrm{vod}}=\frac{\lambda L_{\mathrm{e}} w_{\mathrm{v}}^{2.5}\rho}{2.82\varphi^{0.5}W^{0.5}}\ \mathrm{N/m^2}. \tag{10} \]

Similarly, for a single water conduit,

\[ H_{\mathrm{vod}}=\frac{\lambda L_{\mathrm{e}} w_{\mathrm{v}}^{2.5}\rho}{4\varphi W^{0.5}}\ \mathrm{N/m^2}. \tag{11} \]

In formulas (10) and (11),

\[ \begin{aligned} L_{\mathrm{e}}&=\beta L_0 &&\text{is the equivalent length of the water conduit, m;}\\ L_0& &&\text{is the actual length of the water conduit, m;}\\ \beta& &&\text{is the coefficient of local losses;}\\ w_{\mathrm{v}}& &&\text{is the velocity of water in the water conduits, m/s.} \end{aligned} \]

The magnitude of the water velocity in the water conduits depends on local specific conditions and may be determined by the formula

\[ w_{\mathrm{v}}=\sqrt[3]{\frac{202}{\dfrac{4.187R_{\mathrm{n}}A}{\eta_{\mathrm{k}}Q_{\mathrm{p}}^{\prime\prime}}+31}}\ \mathrm{m/s}. \]

Thus, the resistance of the water conduit also depends on the terminal pressure, since the value \(W\) enters formulas (10) and (12).

The total pump head for a double water conduit is

\[ H_{\mathrm{tsn}}=H_{\mathrm{k}}+H_{\mathrm{vod}}+H_{\mathrm{g}} =z\psi \left(\frac{W}{W_{\mathrm{p}}}\right)^2 w_{\mathrm{k}}^2 +\frac{\lambda \beta L_0 \rho w_{\mathrm{v}}^{2.5}}{2.82\varphi^{0.5}W^{0.5}} +9807H_{\mathrm{g}}\ \mathrm{N/m^2}. \tag{12} \]

For a single water conduit, the formula will be analogous to this one, with the only difference that the coefficient in the denominator of the second term will be 4.

Then the annual electricity consumption, in MJ, for pumping circulating water, after substituting into formula (4) the value from (12), is

\[ E_{\mathrm{dv}}= \frac{3.6\cdot 10^{-3}\varphi WA}{\eta_{\mathrm{n}}} \left[ z\psi \left(\frac{W}{W_{\mathrm{p}}}\right)^2 w_{\mathrm{k}}^2 +\frac{\lambda \beta L_0 \rho w_{\mathrm{v}}^{2.5}}{2.82\varphi^{0.5}W^{0.5}} +9807H_{\mathrm{g}} \right]. \tag{13} \]

\(^1\) The indicated statement is valid only for the case under consideration, when the condenser is specified.

The total annual expenditure in rubles for pumping cooling water under the fuel-cost item is

\[ S_m^{\mathrm{var}}=\frac{0.001\,E_{\mathrm{cw}}R_m}{\eta_k Q_p^{n}} \ \text{rub./yr.} \tag{14} \]

Here

\[ \eta_k=\eta_{ku}\cdot \eta_{\mathrm{pot}}\cdot f_{\mathrm{khoz}}\cdot \eta_t^{p}\cdot \eta_{oi}\cdot \eta_{\mathrm{mech}}\cdot \eta_{\mathrm{gen}}, \]

where \(\eta_t^{p}\) is the thermal efficiency of the cycle of the steam-turbine installation.

Knowing the magnitudes and parameters of the steam extractions for regeneration, \(\eta_t^{p}\) can be calculated as for a regenerative cycle. But with a sufficient degree of accuracy one may take \(\eta_t^{p}=k_p\eta_t\), where \(k_p=1 \div 1.12\) accounts for the increase in thermal efficiency due to regenerative heating of the feedwater.

As is evident from the preceding, when the terminal pressure changes, the quantity of cooling water will change. Consequently, the diameters of the circulating water conduits and the capacities of the circulating pumps will change. The latter will cause a change in production costs depending on capital expenditures for the water conduits and the pumping station. The capital expenditures for the water conduits and pumping station may be determined as follows. The cost of one running meter of water conduit as laid is

\[ k_{\mathrm{vod}}=ad_v \ \text{rub./m}, \tag{15} \]

where the coefficient \(a\) depends on the number of water-conduit lines. If a double water conduit is assumed, then \(a_2=220\) rub./m; if a single one, then \(a_1=110\) rub./m. Then the sum of the annual charges for depreciation and current repair, at a rate of charges \(u_v\), will be, for the water conduit,

\[ S_k^{\mathrm{vod}}=0.01u_vL_0ad_v \ \text{rub./yr.} \tag{16} \]

If, instead of \(d_v\), its value from equations (8) and (9) is substituted, then for a double water conduit we obtain

\[ S_k^{\mathrm{vod}}= \frac{1.414\cdot 10^{-2}u_vL_0a_2\varphi^{0.5}W^{0.5}} {w_v^{0.5}} \ \text{rub./yr.} \tag{17} \]

For a single water conduit, the charges for depreciation and current repair will be

\[ S_k^{\mathrm{vod}}= \frac{2\cdot 10^{-2}u_vL_0a_1\varphi^{0.5}W^{0.5}} {w_v^{0.5}} \ \text{rub./yr.} \tag{18} \]

Here \(L_0\) is the length of the circulating water conduit. If the route of the cooling-water conduits is unknown, then for \(L_0\) one may take the distance of the station from the water-supply source. The cost of the pumping station is

\[ K_n=k_nN_{\mathrm{cn}}=k_cN_{\mathrm{cn}}+k_oN_{\mathrm{cn}}. \tag{19} \]

Here \(N_{\mathrm{cn}}\) is the capacity of the circulating pumps, kW;
\(k_c\) is the cost of the civil-engineering part, rub./kW;
\(k_o\) is the cost of the pumping-station equipment, rub./kW.

The division of the cost of the pumping station into the civil-engineering part and equipment is caused by the different percentages of charges for depreciation and current repair for the equipment, \(u_o\), and for the civil-engineering part, \(u_c\). The charges for the pumping station are

\[ S_k^{\mathrm{nas}}= 0.01\,(u_ck_c+u_ok_o)\frac{E_{\mathrm{cw}}}{A} \ \text{rub./yr.} \tag{20} \]

Thus, we have considered the components of the expenditures for the turbine installation that depend on \(P_k\).

The total annual expenditure in rubles for pumping the cooling water under the fuel-cost item is

\[ S_{m}^{\mathrm{per}}=\frac{0.001\,\mathcal{E}_{\mathrm{ev}}R_{m}}{\eta_{k}Q_{p}^{n}} \ \text{rub/year}. \tag{14} \]

Here

\[ \eta_{k}=\eta_{ku}\eta_{\mathrm{pot}}\eta_{\mathrm{khoz}}\eta_{t}^{p}\eta_{oi}\eta_{\mathrm{mekh}}\eta_{\mathrm{gen}}, \]

where \(\eta_{t}^{p}\) is the thermal efficiency of the cycle of the steam-turbine installation.

Knowing the quantities and parameters of the steam extractions for regeneration, \(\eta_{t}^{p}\) can be calculated as for the regenerative cycle. But with sufficient accuracy one may take \(\eta_{t}^{p}=k_{p}\eta_{t}\), where \(k_{p}=1 \div 1.12\) accounts for the increase in thermal efficiency due to regenerative heating of feedwater.

As is evident from the preceding, with a change in the terminal pressure the quantity of cooling water will change. Consequently, the diameters of the circulating water conduits and the capacities of the circulating pumps will change. The latter will determine the change in production costs depending on capital expenditures for the water conduits and the pump house. The capital expenditures for the water conduits and the pump house can be determined as follows. The cost of one running meter of water conduit as laid is

\[ k_{\mathrm{vod}}=ad_{\mathrm{v}} \ \text{rub/r.m.}, \tag{15} \]

where the coefficient \(a\) depends on the number of conduit lines. If it is proposed to build a double water conduit, then \(a_{2}=220\ \text{rub/m}\); if a single one, then \(a_{1}=110\ \text{rub/m}\). Then the sum of annual charges for depreciation and current repairs, at the charge rate \(u_{\mathrm{v}}\), will be, for the water conduit,

\[ S_{k}^{\mathrm{vod}}=0.01u_{\mathrm{v}}L_{0}ad_{\mathrm{v}}\ \text{rub/year}. \tag{16} \]

If, instead of \(d_{\mathrm{v}}\), its value from equations (8) and (9) is substituted, then for a double water conduit we obtain

\[ S_{k}^{\mathrm{vod}}=\frac{1.414\cdot 10^{-2}u_{\mathrm{v}}L_{0}a_{2}\varphi^{0.5}W^{0.5}}{w_{\mathrm{v}}^{0.5}}\ \text{rub/year}. \tag{17} \]

For a single water conduit, the charges for depreciation and current repairs will be

\[ S_{k}^{\mathrm{vod}}=\frac{2\cdot 10^{-2}u_{\mathrm{v}}L_{0}a_{1}\varphi^{0.5}W^{0.5}}{w_{\mathrm{v}}^{0.5}}\ \text{rub/year}. \tag{18} \]

Here \(L_{0}\) is the length of the circulating water conduit. If the route of the cooling-water conduits is unknown, then for \(L_{0}\) one may take the distance of the station from the source of water supply. The cost of the pump house is

\[ K_{\mathrm{n}}=\kappa_{\mathrm{n}}N_{\mathrm{tsn}}=\kappa_{c}N_{\mathrm{tsn}}+\kappa_{0}N_{\mathrm{tsn}}. \tag{19} \]

Here \(N_{\mathrm{tsn}}\) is the capacity of the circulating pumps, kW;

\[ \kappa_{c} \]

is the cost of the structural part, rub/kW;

\[ \kappa_{0} \]

is the cost of the pump-house equipment, rub/kW.

The division of the cost of the pump house into the structural part and the equipment is caused by the different percentages of charges for depreciation and current repairs for the equipment, \(u_{0}\), and for the structural part, \(u_{c}\). The charges for the pump house are

\[ S_{k}^{\mathrm{nas}}=0.01\left(u_{c}\kappa_{c}+u_{0}\kappa_{0}\right)\frac{\mathcal{E}_{\mathrm{ev}}}{A}\ \text{rub/year}. \tag{20} \]

Thus, we have considered the components of the expenses for the turbine installation that depend on \(P_{k}\).

![Figure 1 graph]

Fig. 1. Dependence of the variable production costs of the unit on the final pressure for different fuel costs.

![Figure 2 graph]

Fig. 2. Dependence of the optimal final pressure on the cost of fuel.

To these expenses one must add a certain percentage of the capital costs, in accordance with the adopted efficiency coefficient of capital expenditures \(\delta\),

\[ \delta(\kappa_{\mathrm{v}}L_0+\kappa_{\mathrm{n}}N_{\mathrm{tsn}}). \]

Here \(\kappa_{\mathrm{v}}\) and \(\kappa_{\mathrm{n}}\) are determined from formulas (15) and (19).

In Fig. 1, using this method, dependences have been plotted of the annual monetary expenditures for the installation as functions of \(P_k\) for the particular case in which

\[ N_{\mathrm{e}}=50000\ \text{kW};\quad P_0=90\ \text{ata};\quad t_0=535^\circ\text{C}; \]

\[ t_1=10^\circ\text{C};\quad z=2;\quad Hz=10\ \text{m};\quad L_0=500\ \text{m}; \]

\[ t_s-t_2=5^\circ\text{C} \]

for various fuel costs.

From Fig. 1 the following conclusions may be drawn:

1. With a decrease in the cost of the fuel being burned, a higher final pressure should be adopted when designing water-supply systems.

2. The cheaper the fuel, the flatter the curve of the dependence \(\Sigma S=f(P_k)\), and the less perceptible the deviation of the final pressure \(P_k\) from the optimum \(P_k^{\mathrm{opt}}\).

3. Since the left branch of the curves rises more steeply, deviation of \(P_k\) from the optimum toward a deeper vacuum will lead to more substantial installation costs in comparison with the optimum \(P_k^{\mathrm{opt}}\).

In Fig. 2 the dependence \(P_k^{\mathrm{opt}}=f(R_m)\) is given for the same conditions. The dependence shows that, at a fuel cost of 5 rubles/t and higher and under favorable water-supply conditions, \(P_k^{\mathrm{opt}}\) depends only weakly on \(R_m\) and may be taken equal to the calculated value. At a cost \(R_m<5\) rubles/t, it is necessary in each specific case to determine \(P_k^{\mathrm{opt}}\) and to calculate and design the circulating water-supply system for this pressure.